An astronaut in orbit floats even though Earth is still pulling strongly. The spacecraft and astronaut fall together while continually missing the ground. That picture connects an everyday falling object to a circular orbit.
We will also combine several sources and examine the interior of a uniform sphere. These cases expose two common errors: adding fields as scalars and using a whole-body point-mass formula inside the source.
What you will learn to do
- Use centre distance in an external spherical gravitational field.
- Add source fields as vectors and potentials as scalars.
- Derive circular speed, period and energy from stated assumptions.
- Distinguish orbital free fall from absence of gravity.
- Calculate escape and asymptotic speeds from energy.
- Derive interior field and potential for a uniform sphere and check the surface match.
- Connect the three Kepler laws to ellipse geometry, angular momentum and energy.
- Distinguish a geosynchronous period from geostationary geometry.
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Gravitational fields, orbits and escape Learning goals: - Use centre distance in an external spherical gravitational field. - Add source fields as vectors and potentials as scalars. - Derive circular speed, period and energy from stated assumptions. - Distinguish orbital free fall from absence of gravity. - Calculate escape and asymptotic speeds from energy. - Derive interior field and potential for a uniform sphere and check the surface match. - Connect the three Kepler laws to ellipse geometry, angular momentum and energy. - Distinguish a geosynchronous period from geostationary geometry. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
BEFORE YOU READ
Bring back one idea
Try a sentence or a sketch before opening the comparison. If this is new, use the explanation to get started.
What interaction supplies the inward acceleration of a satellite in a simple circular Earth orbit?
Compare your reasoning
Gravity supplies it. There is no extra inward force beyond gravity in this model. The required acceleration is v²/r, where r is measured from Earth’s centre.
Revisit the explanation →01 / Gravitational fields, orbits and escape
Gravity is a field, and distance is measured from the source
Newtonian gravity between two point masses has magnitude GMm/r² and acts along the line joining them, attractively. Divide the force on a small test mass by that mass to obtain gravitational field g. The field describes the acceleration a freely falling test particle would have under gravity alone in the model.
For an isolated spherically symmetric body, the field outside its surface is the same as if its total mass were concentrated at its centre. The distance r is therefore measured from the centre, not from the surface. At altitude h above a spherical body of radius R, use r = R + h.
The constant-G Newtonian model applies to the ordinary weak-field, low-speed situations considered here. We also neglect atmosphere, other astronomical bodies and the source’s response when treating a much smaller satellite around a fixed primary. These assumptions should be revisited for precision or comparable-mass problems.
The radial unit vector points outward from the source, so the minus sign makes the field inward. G has units m³/(kg·s²).
WORKED EXAMPLE
Altitude is not the inverse-square distance
An ideal spherical planet has surface gravitational field 8 m/s² and radius R. Find its field magnitude at altitude R above the surface.
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Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Gravitational fields, orbits and escape
Focus: Gravity is a field, and distance is measured from the source
Learning goals:
- Use centre distance in an external spherical gravitational field.
- Add source fields as vectors and potentials as scalars.
- Derive circular speed, period and energy from stated assumptions.
- Distinguish orbital free fall from absence of gravity.
- Calculate escape and asymptotic speeds from energy.
- Derive interior field and potential for a uniform sphere and check the surface match.
- Connect the three Kepler laws to ellipse geometry, angular momentum and energy.
- Distinguish a geosynchronous period from geostationary geometry.
Equations in this section (LaTeX):
\mathbf g(r)=-\frac{GM}{r^2}\hat{\mathbf r},\qquad\mathbf F=m\mathbf g,\qquad r=R+h\quad(r\geq R)
Meaning: The radial unit vector points outward from the source, so the minus sign makes the field inward. G has units m³/(kg·s²).
Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.
Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.02 / Gravitational fields, orbits and escape
Add gravitational fields as vectors
Every source contributes its own field at the observation point. The total field is the vector sum of those contributions. Compute the direction toward each mass and resolve components before adding. Summing positive magnitudes is only valid when the fields point in the same direction.
For two positive masses on a line, fields oppose between the masses and point in the same direction on either exterior side. A zero-field point can lie between them, closer to the smaller mass because a smaller distance compensates for its weaker source strength. It does not lie outside the pair in this simple two-source model.
For a continuous distribution, add small mass-element contributions through an integral. Symmetry can cancel components, but it must be demonstrated from the actual distribution. A point at the geometric centre of an asymmetric mass arrangement does not automatically have zero field.
Each numerator points from the observation point toward a source. The cubic denominator combines direction normalisation with inverse-square magnitude.
WORKED EXAMPLE
Find the zero field between unequal sources
Two point masses M and 4M are fixed a distance 3a apart. Find the zero-field point along the line joining them, measured from M.
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STOP & THINK
Why can the field of two positive point masses not cancel at a point on the line outside both masses?
- Because both fields point toward the pair in the same direction
- Because gravitational fields cannot ever cancel
- Because the larger mass screens the smaller one
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Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Gravitational fields, orbits and escape
Focus: Add gravitational fields as vectors
Learning goals:
- Use centre distance in an external spherical gravitational field.
- Add source fields as vectors and potentials as scalars.
- Derive circular speed, period and energy from stated assumptions.
- Distinguish orbital free fall from absence of gravity.
- Calculate escape and asymptotic speeds from energy.
- Derive interior field and potential for a uniform sphere and check the surface match.
- Connect the three Kepler laws to ellipse geometry, angular momentum and energy.
- Distinguish a geosynchronous period from geostationary geometry.
Equations in this section (LaTeX):
\mathbf g(\mathbf r)=\sum_i\frac{GM_i(\mathbf r_i-\mathbf r)}{|\mathbf r_i-\mathbf r|^3}
Meaning: Each numerator points from the observation point toward a source. The cubic denominator combines direction normalisation with inverse-square magnitude.
Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.
Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.03 / Gravitational fields, orbits and escape
Gravitational potential adds as a scalar
With the gravitational potential chosen to approach zero at infinity, a point source has potential Φ = −GM/r and a test mass has potential energy U = mΦ. The negative value indicates a bound state relative to separated bodies at rest at infinity. It does not represent a negative field magnitude or an impossible energy.
Potential contributions add algebraically as scalars. Field contributions add as vectors. All positive gravitational point sources give negative contributions to Φ with this reference, even where their field vectors cancel. A zero-field point can therefore have nonzero potential.
The field is the negative gradient of potential. In a radial situation, gr = −dΦ/dr. Differentiating −GM/r gives +GM/r², and the extra negative sign gives the inward field component. Near a surface over a small height change, the exact potential difference reduces approximately to gΔh per unit mass; that constant-g approximation is not valid for arbitrary large altitude changes.
Potential has units J/kg; gravitational field has units N/kg or m/s². Their values answer different questions.
WORKED EXAMPLE
A field cancellation does not cancel the potential
For the two sources M and 4M separated by 3a, calculate the potential at the zero-field point a from M and 2a from 4M. Use zero potential at infinity.
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Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Gravitational fields, orbits and escape
Focus: Gravitational potential adds as a scalar
Learning goals:
- Use centre distance in an external spherical gravitational field.
- Add source fields as vectors and potentials as scalars.
- Derive circular speed, period and energy from stated assumptions.
- Distinguish orbital free fall from absence of gravity.
- Calculate escape and asymptotic speeds from energy.
- Derive interior field and potential for a uniform sphere and check the surface match.
- Connect the three Kepler laws to ellipse geometry, angular momentum and energy.
- Distinguish a geosynchronous period from geostationary geometry.
Equations in this section (LaTeX):
\Phi=-\frac{GM}{r},\qquad U=m\Phi,\qquad\Delta U=GMm\left(\frac1{r_i}-\frac1{r_f}\right)
Meaning: Potential has units J/kg; gravitational field has units N/kg or m/s². Their values answer different questions.
Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.
Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.04 / Gravitational fields, orbits and escape
Gravity supplies the inward acceleration of a circular orbit
A satellite in a circular orbit is continually falling toward its primary while its sideways velocity carries it around. In the ideal two-body approximation with a much heavier spherical primary, gravity provides the entire inward resultant. Set GMm/r² = mv²/r and cancel the satellite mass.
The resulting circular speed is √(GM/r). It is a condition for a circular orbit at that radius, not the speed of every object passing through that radius. An eccentric orbit can have a different speed and a radial velocity component. A satellite given the circular speed in a non-tangential direction does not automatically follow a circle.
The period is circumference divided by speed: T = 2π√(r³/GM). Thus T² is proportional to r³ for circular orbits around the same dominant source. A larger circular orbit has a lower speed but a longer path and a longer period.
The orbit radius is centre-to-centre distance. Satellite mass cancels when its effect on the primary’s motion is neglected.
WORKED EXAMPLE
Compute an orbit with a stated gravitational parameter
An ideal spherical planet has GM = 8.0 × 10¹² m³/s². A small satellite follows a circular orbit of radius 2.0 × 10⁶ m from its centre. Find orbital speed and period.
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STOP & THINK
Two circular orbits around the same dominant mass have radii r and 4r. What is the ratio T(4r)/T(r)?
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Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Gravitational fields, orbits and escape
Focus: Gravity supplies the inward acceleration of a circular orbit
Learning goals:
- Use centre distance in an external spherical gravitational field.
- Add source fields as vectors and potentials as scalars.
- Derive circular speed, period and energy from stated assumptions.
- Distinguish orbital free fall from absence of gravity.
- Calculate escape and asymptotic speeds from energy.
- Derive interior field and potential for a uniform sphere and check the surface match.
- Connect the three Kepler laws to ellipse geometry, angular momentum and energy.
- Distinguish a geosynchronous period from geostationary geometry.
Equations in this section (LaTeX):
v_c=\sqrt{\frac{GM}{r}},\qquad T=2\pi\sqrt{\frac{r^3}{GM}}
Meaning: The orbit radius is centre-to-centre distance. Satellite mass cancels when its effect on the primary’s motion is neglected.
Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.
Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.05 / Gravitational fields, orbits and escape
A higher circular orbit is slower but has greater total energy
For a circular orbit, kinetic energy is GMm/(2r), potential energy is −GMm/r and total mechanical energy is −GMm/(2r). The potential-energy magnitude is twice the kinetic energy. Their sum is negative with the infinity reference.
Increasing the circular orbit radius makes the speed and kinetic energy smaller, while potential energy rises by a larger amount. Total energy therefore increases—it becomes less negative. Reaching a higher circular orbit requires an overall energy input even though the final circular speed is lower.
An instantaneous speed change at one point does not generally move a satellite directly to another circular orbit. It usually creates a different, often elliptical trajectory unless additional manoeuvres establish the required radius and tangential speed. Comparing endpoint circular energies gives the total energy difference, not a complete trajectory or fuel calculation.
WORKED EXAMPLE
Compare two circular energy states per kilogram
Around the ideal planet with GM = 8.0 × 10¹² m³/s², compare circular orbits of radii 1.0 × 10⁶ m and 2.0 × 10⁶ m. Find the increase in total mechanical energy per kilogram. Treat both radii as outside the body and ignore any atmosphere.
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Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Gravitational fields, orbits and escape Focus: A higher circular orbit is slower but has greater total energy Learning goals: - Use centre distance in an external spherical gravitational field. - Add source fields as vectors and potentials as scalars. - Derive circular speed, period and energy from stated assumptions. - Distinguish orbital free fall from absence of gravity. - Calculate escape and asymptotic speeds from energy. - Derive interior field and potential for a uniform sphere and check the surface match. - Connect the three Kepler laws to ellipse geometry, angular momentum and energy. - Distinguish a geosynchronous period from geostationary geometry. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
06 / Gravitational fields, orbits and escape
An orbiting scale reads support rather than gravity
An astronaut and an orbiting cabin accelerate under nearly the same gravitational field. In a sufficiently small freely falling cabin, a resting astronaut does not need a floor force to share that acceleration. A scale can read zero even though the gravitational field is substantial.
This is apparent weightlessness: the support force is absent. It is not the absence of gravity, and it does not require being far enough away for gravity to vanish. In a circular orbit gravity is precisely the interaction continuously turning the motion.
For a satellite to remain above one point on an ideal rotating planet, it must share the planet’s rotation period and direction and orbit in the equatorial plane. Equal period alone gives a synchronous orbit, but an inclined or eccentric orbit will not stay over the same surface location. The simple stationary-above-a-point case is circular and equatorial.
STOP & THINK
Why can an astronaut’s scale read zero in a low orbit where gravity is still strong?
- Gravity is cancelled by a new outward force in every frame
- The astronaut and cabin are in shared free fall, so support can be zero
- The astronaut has lost mass
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Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Gravitational fields, orbits and escape Focus: An orbiting scale reads support rather than gravity Learning goals: - Use centre distance in an external spherical gravitational field. - Add source fields as vectors and potentials as scalars. - Derive circular speed, period and energy from stated assumptions. - Distinguish orbital free fall from absence of gravity. - Calculate escape and asymptotic speeds from energy. - Derive interior field and potential for a uniform sphere and check the surface match. - Connect the three Kepler laws to ellipse geometry, angular momentum and energy. - Distinguish a geosynchronous period from geostationary geometry. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
07 / Gravitational fields, orbits and escape
Escape speed follows from the energy needed to reach infinity
Escape from an isolated spherical body does not mean reaching a place where gravity suddenly switches off. The field decreases continuously with distance. The minimum outward launch energy for an unobstructed escape is the energy that leaves the particle with speed approaching zero as distance approaches infinity.
With zero potential at infinity, this limiting total mechanical energy is zero. Set ½mv² − GMm/r = 0 to obtain vesc = √(2GM/r). The test mass cancels. The result assumes no atmosphere, no additional propulsion after launch, a dominant fixed source and an outward trajectory that avoids the body.
At the same radius, escape speed is √2 times circular speed. These are two different initial-state conditions. A satellite already in a circular orbit has kinetic energy; the additional tangential speed needed for marginal escape is vesc − vc in the impulsive idealisation, not another full escape speed added to its existing velocity.
This expression applies outside a spherical source, with the infinity potential reference and the stated isolated-source model.
WORKED EXAMPLE
Find the marginal escape speed from a model surface
An ideal spherical planet has GM = 8.0 × 10¹² m³/s² and radius 1.0 × 10⁶ m. Neglect rotation and atmosphere. Find the outward surface escape speed.
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STOP & THINK
In the isolated Newtonian escape-speed formula, why does the escaping particle’s mass not affect the threshold speed?
- Gravity does not act on heavier particles
- Its mass multiplies both kinetic and potential energy and cancels
- All particles have the same kinetic energy
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Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Gravitational fields, orbits and escape
Focus: Escape speed follows from the energy needed to reach infinity
Learning goals:
- Use centre distance in an external spherical gravitational field.
- Add source fields as vectors and potentials as scalars.
- Derive circular speed, period and energy from stated assumptions.
- Distinguish orbital free fall from absence of gravity.
- Calculate escape and asymptotic speeds from energy.
- Derive interior field and potential for a uniform sphere and check the surface match.
- Connect the three Kepler laws to ellipse geometry, angular momentum and energy.
- Distinguish a geosynchronous period from geostationary geometry.
Equations in this section (LaTeX):
v_{\mathrm{esc}}=\sqrt{\frac{2GM}{r}}=\sqrt2\,v_c
Meaning: This expression applies outside a spherical source, with the infinity potential reference and the stated isolated-source model.
Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.
Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.08 / Gravitational fields, orbits and escape
A speed above escape leaves kinetic energy at large distance
If the launch speed exceeds the escape threshold and the trajectory remains unobstructed, the particle retains nonzero speed far from the source. Equating its initial energy with ½mv∞² gives v∞² = v₀² − vesc². The speed excess is not found by subtracting speeds linearly because kinetic energy is quadratic.
If v₀ is below escape speed for a radial outward launch, the total energy is negative and there is a finite turning radius. Set the speed to zero at that radius and solve the same energy equation. This determines the farthest radius, not the elapsed travel time.
A negative total energy can also describe a bound elliptical orbit when angular momentum is nonzero. A positive total energy does not prevent a trajectory aimed into the source from striking its surface. Energy describes accessibility; direction and angular momentum describe the path. Keep those roles separate.
WORKED EXAMPLE
Subtract energy contributions, not speed magnitudes
A particle is launched radially outward from the model planet’s surface at 5.0 km/s, where the escape speed is 4.0 km/s. With no further propulsion or losses, find its speed as distance becomes very large.
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STOP & THINK
An outward launch has speed 13 km/s at a point where escape speed is 12 km/s. What is the asymptotic speed in the isolated lossless model?
Answer in km/s.
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Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Gravitational fields, orbits and escape Focus: A speed above escape leaves kinetic energy at large distance Learning goals: - Use centre distance in an external spherical gravitational field. - Add source fields as vectors and potentials as scalars. - Derive circular speed, period and energy from stated assumptions. - Distinguish orbital free fall from absence of gravity. - Calculate escape and asymptotic speeds from energy. - Derive interior field and potential for a uniform sphere and check the surface match. - Connect the three Kepler laws to ellipse geometry, angular momentum and energy. - Distinguish a geosynchronous period from geostationary geometry. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
09 / Gravitational fields, orbits and escape
Inside a spherical mass distribution, only enclosed mass sets the field
Newton’s shell theorem states that a spherically symmetric shell produces no net gravitational field anywhere inside that shell and acts externally like a point mass at its centre. For a spherical body, split the source into shells. At an interior radius r, shells outside r contribute zero field, while the enclosed mass contributes as a central source.
Thus the interior field magnitude is GMenc(r)/r². For a uniform-density sphere, Menc = M(r/R)³ because enclosed mass is proportional to enclosed volume. Substitution gives g(r) = GMr/R³: the field rises linearly from zero at the centre to GM/R² at the surface.
Using GM/r² with the entire mass at an interior point would wrongly predict a divergence at the centre. That expression treats all source mass as interior, which is only valid outside the body. A nonuniform spherical density still permits the enclosed-mass field formula, but Menc(r) need not scale as r³ and the field need not be linear.
The last expression requires uniform density. At r = 0, use its continuous limit g = 0 rather than substituting into a 0/0 form.
Follow gravity from the centre through the surface
- Field magnitude
- 0
WORKED EXAMPLE
The same field magnitude can occur at two radii
A uniform-density spherical planet has surface field 8 m/s². Find the field magnitude at r = R/2 and identify the exterior radius where the same magnitude occurs.
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Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Gravitational fields, orbits and escape
Focus: Inside a spherical mass distribution, only enclosed mass sets the field
Learning goals:
- Use centre distance in an external spherical gravitational field.
- Add source fields as vectors and potentials as scalars.
- Derive circular speed, period and energy from stated assumptions.
- Distinguish orbital free fall from absence of gravity.
- Calculate escape and asymptotic speeds from energy.
- Derive interior field and potential for a uniform sphere and check the surface match.
- Connect the three Kepler laws to ellipse geometry, angular momentum and energy.
- Distinguish a geosynchronous period from geostationary geometry.
Equations in this section (LaTeX):
g(r)=\frac{GM_{\mathrm{enc}}(r)}{r^2},\qquad M_{\mathrm{enc}}=M\frac{r^3}{R^3}\quad\Longrightarrow\quad g(r)=\frac{GM}{R^3}r\quad(0\leq r\leq R)
Meaning: The last expression requires uniform density. At r = 0, use its continuous limit g = 0 rather than substituting into a 0/0 form.
Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.
Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.10 / Gravitational fields, orbits and escape
Zero field inside a shell does not mean zero potential
A thin spherical shell has constant potential −GM/R throughout its interior when zero is chosen at infinity. The field there is zero because the potential has no spatial slope, not because its value is zero. Work is required to escape from inside the shell even though a particle can move within the empty interior without a gravitational force from that shell.
For a uniform solid sphere, obtain the interior potential by integrating the interior field and matching the surface value −GM/R. This gives Φ(r) = −GM(3R² − r²)/(2R³). Its derivative reproduces the inward linear field, and it joins the exterior potential continuously at the surface.
At the centre, potential is −3GM/(2R), more negative than at the surface, while the field is zero by symmetry. A particle at rest at the exact centre has no gravitational acceleration, but still has negative energy relative to infinity. Local force and total escape-energy requirement are distinct questions.
Uniform density and spherical symmetry are required. The potential reference remains zero at infinity.
WORKED EXAMPLE
A zero local field can coexist with a nonzero escape threshold
Consider an ideal uniform sphere of mass M and radius R. Imagine a negligibly narrow unobstructed radial passage from the centre to the surface, whose removed mass does not significantly change the field. Ignoring collisions and all losses, compare the outward escape speed from the centre with the escape speed from the surface.
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Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Gravitational fields, orbits and escape
Focus: Zero field inside a shell does not mean zero potential
Learning goals:
- Use centre distance in an external spherical gravitational field.
- Add source fields as vectors and potentials as scalars.
- Derive circular speed, period and energy from stated assumptions.
- Distinguish orbital free fall from absence of gravity.
- Calculate escape and asymptotic speeds from energy.
- Derive interior field and potential for a uniform sphere and check the surface match.
- Connect the three Kepler laws to ellipse geometry, angular momentum and energy.
- Distinguish a geosynchronous period from geostationary geometry.
Equations in this section (LaTeX):
\Phi_{\mathrm{inside}}(r)=-\frac{GM}{2R}\left(3-\frac{r^2}{R^2}\right),\qquad\Phi(0)=-\frac{3GM}{2R}
Meaning: Uniform density and spherical symmetry are required. The potential reference remains zero at infinity.
Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.
Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.11 / Gravitational fields, orbits and escape
Choose the correct symmetry and endpoint conditions
Before using a gravitational formula, identify whether the point is outside or inside the source, whether the source is spherical, whether density is uniform and whether a single primary dominates. These choices determine which simplifications are justified.
Next distinguish the quantity requested. Field and force are vectors. Potential and potential energy are scalars. Circular speed comes from a radial force balance; escape speed comes from an endpoint energy condition. A common square-root appearance does not make the formulas interchangeable.
Finally test a limit. A far-away field should approach zero. An interior uniform-sphere field should approach zero at the centre rather than diverge. Circular period should grow with radius. Potential and field should match across a uniform-sphere surface without a jump in their values. These checks can reveal a wrong radius or an inappropriate mass formula before any calculator digits are trusted.
STOP & THINK
At the centre of an isolated uniform solid sphere, with zero potential at infinity, which statement is correct?
- Field and potential are both zero
- Field is zero and potential is negative
- Field is infinite and potential is zero
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Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Gravitational fields, orbits and escape Focus: Choose the correct symmetry and endpoint conditions Learning goals: - Use centre distance in an external spherical gravitational field. - Add source fields as vectors and potentials as scalars. - Derive circular speed, period and energy from stated assumptions. - Distinguish orbital free fall from absence of gravity. - Calculate escape and asymptotic speeds from energy. - Derive interior field and potential for a uniform sphere and check the surface match. - Connect the three Kepler laws to ellipse geometry, angular momentum and energy. - Distinguish a geosynchronous period from geostationary geometry. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
12 / Gravitational fields, orbits and escape
An ellipse needs a size, a shape and a focus
The circular model is a useful special case, but an orbiting body need not stay at one distance. For a bound Kepler orbit in the ideal inverse-square two-body model, its path is an ellipse with the attracting focus at the primary, when the primary is much more massive. In the exact two-body description, the relative separation follows the ellipse and the bodies orbit their common centre of mass. Here we use μ = GM for a dominant primary and neglect drag, thrust, other bodies and relativistic corrections.
The semimajor axis a is half the longest diameter of the ellipse. Its eccentricity e describes the shape, with 0≤e<1 for a nondegenerate bound ellipse. The nearest and farthest distances from the occupied focus are rₚ = a(1−e) and rₐ = a(1+e). They are distances from the primary centre, not heights above its surface. Adding the two removes e; subtracting them isolates the shape.
A circle has e=0 and rₚ=rₐ=a. Giving a body the local circular speed in a non-tangential direction does not make its orbit circular: the direction and radial component matter too. An orbital sketch should identify the occupied focus, not place the primary at the ellipse centre by habit.
The sum sets the orbit size and the difference sets its eccentricity.
WORKED EXAMPLE
Recover an ellipse from its two extreme distances
A model satellite remains between 10,000 km and 30,000 km from its primary centre. Find a and e.
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Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Gravitational fields, orbits and escape
Focus: An ellipse needs a size, a shape and a focus
Learning goals:
- Use centre distance in an external spherical gravitational field.
- Add source fields as vectors and potentials as scalars.
- Derive circular speed, period and energy from stated assumptions.
- Distinguish orbital free fall from absence of gravity.
- Calculate escape and asymptotic speeds from energy.
- Derive interior field and potential for a uniform sphere and check the surface match.
- Connect the three Kepler laws to ellipse geometry, angular momentum and energy.
- Distinguish a geosynchronous period from geostationary geometry.
Equations in this section (LaTeX):
a=\frac{r_p+r_a}{2},\qquad e=\frac{r_a-r_p}{r_a+r_p}
Meaning: The sum sets the orbit size and the difference sets its eccentricity.
Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.
Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.13 / Gravitational fields, orbits and escape
Equal swept areas connect orbital geometry to angular momentum
A central gravitational force has zero torque about the attracting focus. The satellite’s angular momentum about that focus is therefore constant in this model. In a short time interval the swept sector has area approximately ½r v⊥ Δt, where v⊥ is the velocity component perpendicular to the radius. Taking the short-time limit gives dA/dt = ½r v⊥ = L/(2m). This is Kepler’s area law.
Equal areas do not mean equal path lengths or equal angles. Far from the primary, a shorter arc can sweep the same area because its radius is larger. Near the primary the body moves faster. The relation r v = constant applies at the two turning points because the radial velocity is zero there; at an arbitrary point use the transverse component rather than the entire speed.
L is angular momentum magnitude about the primary and m is the orbiting mass. The second relation compares the two turning points, where velocity is purely tangential.
WORKED EXAMPLE
A speed ratio and an angular-speed ratio are different
For the orbit with rₚ = 10,000 km and rₐ = 30,000 km, compare the endpoint speeds and angular speeds.
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STOP & THINK
An ideal elliptical orbit has rₐ/rₚ = 2.5. What is the ratio vₚ/vₐ of endpoint speed magnitudes?
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Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Gravitational fields, orbits and escape
Focus: Equal swept areas connect orbital geometry to angular momentum
Learning goals:
- Use centre distance in an external spherical gravitational field.
- Add source fields as vectors and potentials as scalars.
- Derive circular speed, period and energy from stated assumptions.
- Distinguish orbital free fall from absence of gravity.
- Calculate escape and asymptotic speeds from energy.
- Derive interior field and potential for a uniform sphere and check the surface match.
- Connect the three Kepler laws to ellipse geometry, angular momentum and energy.
- Distinguish a geosynchronous period from geostationary geometry.
Equations in this section (LaTeX):
\frac{dA}{dt}=\frac{L}{2m},\qquad r_pv_p=r_av_a
Meaning: L is angular momentum magnitude about the primary and m is the orbiting mass. The second relation compares the two turning points, where velocity is purely tangential.
Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.
Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.14 / Gravitational fields, orbits and escape
The semimajor axis fixes energy and period, not local speed
The orbital energy per unit mass is ε = v²/2−μ/r. Evaluate it at both turning points and use rₚvₚ=rₐvₐ. Eliminating one endpoint speed gives ε = −μ/(rₚ+rₐ) = −μ/(2a). Putting that energy back into the equation at any point gives v² = μ(2/r−1/a), often called the vis-viva relation. A fixed orbit has constant a and energy, but its local r and speed change.
The period of a bound inverse-square orbit is T = 2π√(a³/μ). One way to connect this to the area law is to divide the ellipse area πab by the constant area rate L/(2m), using b=a√(1−e²) and (L/m)²=μa(1−e²). The shape factors cancel. At fixed a and primary mass, changing eccentricity changes the speed history but not the period.
The exact two-body relative-orbit version uses μ=G(M+m), not GM. Do not compare periods around different primaries using only the semimajor-axis ratio. The same central mass is a condition of the simple ratio rule.
The circular limit has r=a and restores v²=μ/r. Specific energy ε has units J/kg = m²/s².
Explain this equation
Read it in words
Local speed depends on local distance and the size of the whole orbit; the period depends on the semimajor axis.
Symbols and units
- Semimajor axis of the relative orbitm
- Current separation from the primary centre in the dominant-primary approximationm
- Gravitational parameter, approximately GMm³/s²
- Speed relative to the primary in that approximationm/s
- Orbital periods
Signs and operations
The energy of a bound orbit is negative relative to rest at infinite separation. The term 2/r−1/a is positive on the ellipse.
When it applies
Unforced bound two-body inverse-square orbit; use G(M+m) for exact relative motion when neither mass is negligible.
Make it concrete
At r=a, the speed squared becomes μ/a. As eccentricity tends to zero, this applies everywhere.
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Equation (LaTeX): \begin{aligned}\varepsilon&=-\frac{\mu}{2a}\\v^2&=\mu\left(\frac{2}{r}-\frac{1}{a}\right)\\T^2&=\frac{4\pi^2a^3}{\mu}\end{aligned}
Read it as: Local speed depends on local distance and the size of the whole orbit; the period depends on the semimajor axis.
Symbols:
a: Semimajor axis of the relative orbit; m
r: Current separation from the primary centre in the dominant-primary approximation; m
\mu: Gravitational parameter, approximately GM; m³/s²
v: Speed relative to the primary in that approximation; m/s
T: Orbital period; s
Signs and operations: The energy of a bound orbit is negative relative to rest at infinite separation. The term 2/r−1/a is positive on the ellipse.
Conditions: Unforced bound two-body inverse-square orbit; use G(M+m) for exact relative motion when neither mass is negligible.
Help me explain the equation in words, predict how changing a quantity affects the result, and apply it to a small numerical example.
Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.WORKED EXAMPLE
Check the speed against the conserved energy
Use μ = 4.0×10¹⁴ m³/s², a = 2.0×10⁷ m and rₚ = 1.0×10⁷ m. Find the nearest-point speed to three significant figures.
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Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Gravitational fields, orbits and escape
Focus: The semimajor axis fixes energy and period, not local speed
Learning goals:
- Use centre distance in an external spherical gravitational field.
- Add source fields as vectors and potentials as scalars.
- Derive circular speed, period and energy from stated assumptions.
- Distinguish orbital free fall from absence of gravity.
- Calculate escape and asymptotic speeds from energy.
- Derive interior field and potential for a uniform sphere and check the surface match.
- Connect the three Kepler laws to ellipse geometry, angular momentum and energy.
- Distinguish a geosynchronous period from geostationary geometry.
Equations in this section (LaTeX):
\begin{aligned}\varepsilon&=-\frac{\mu}{2a}\\v^2&=\mu\left(\frac{2}{r}-\frac{1}{a}\right)\\T^2&=\frac{4\pi^2a^3}{\mu}\end{aligned}
Meaning: The circular limit has r=a and restores v²=μ/r. Specific energy ε has units J/kg = m²/s².
Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.
Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.15 / Gravitational fields, orbits and escape
Matching the period is necessary, but does not fix the ground position
A satellite that appears fixed above one terrestrial longitude must turn with Earth’s rotation and stay above the equator. In the ideal model its orbit is circular, equatorial and prograde, with period equal to one sidereal rotation. An inclined orbit with that period moves north and south in the sky. An eccentric one changes angular speed. A retrograde orbit moves in the wrong direction even if its period has the same magnitude.
Geosynchronous means the orbital period matches Earth’s rotation period. Geostationary adds the geometry and direction needed for a fixed ground position. Use the sidereal rotation period, about 86,164 s, in the ideal calculation. The 86,400 s mean solar day refers to the Sun returning to a local meridian while Earth also advances around the Sun.
First find the orbital radius from the primary centre, then subtract the planet radius to obtain altitude. A synchronous radius inside the physical body would not be an available free orbit. Real station keeping, non-spherical gravity and perturbations lie outside this introductory model.
Equate the circular orbital period to the rotation period. The cube root produces a length because μT² has units m³.
WORKED EXAMPLE
Turn a rotation period into an altitude
Take Earth-model values μ = 3.986×10¹⁴ m³/s², T_rot = 86,164 s and spherical radius R = 6.371×10⁶ m. Estimate geostationary altitude.
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Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Gravitational fields, orbits and escape
Focus: Matching the period is necessary, but does not fix the ground position
Learning goals:
- Use centre distance in an external spherical gravitational field.
- Add source fields as vectors and potentials as scalars.
- Derive circular speed, period and energy from stated assumptions.
- Distinguish orbital free fall from absence of gravity.
- Calculate escape and asymptotic speeds from energy.
- Derive interior field and potential for a uniform sphere and check the surface match.
- Connect the three Kepler laws to ellipse geometry, angular momentum and energy.
- Distinguish a geosynchronous period from geostationary geometry.
Equations in this section (LaTeX):
r_{\mathrm{sync}}=\left(\frac{\mu T_{\mathrm{rot}}^2}{4\pi^2}\right)^{1/3},\qquad h=r_{\mathrm{sync}}-R_{\mathrm{planet}}
Meaning: Equate the circular orbital period to the rotation period. The cube root produces a length because μT² has units m³.
Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.
Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.BRING THE IDEAS TOGETHER
What should stay with you
- The external field of a spherical source uses its total mass and distance from its centre.
- A zero vector field does not require zero scalar potential.
- Circular motion requires the appropriate tangential speed as well as inward gravity.
- Higher circular orbits are slower but have less negative total energy.
- Escape is an energy threshold under an isolated, lossless model.
- Inside a uniform sphere, enclosed mass produces a field proportional to radius.
- For an ellipse, use a in the period and energy relations and local r in the speed relation.
- A geostationary orbit must be circular, equatorial, prograde and synchronous with sidereal rotation.
RETRIEVE, THEN EXPLAIN
Try it without the examples
Work these out before checking. If an answer surprises you, identify which assumption or step needs another look. A correct choice is most useful when you can explain why the alternatives fail.
SELF CHECK 1
At the surface of a spherical source the field is 12 m/s². What is its magnitude at centre distance 2R, where R is the surface radius?
Answer in m/s².
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SELF CHECK 2
A uniform solid sphere has surface gravitational field 10 m/s². What is the magnitude at r = 0.30R?
Answer in m/s².
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SELF CHECK 3
At a given exterior radius, how do escape and circular speeds compare in the isolated spherical-source model?
- They are equal
- Escape speed is √2 times circular speed
- Escape speed is half circular speed
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SELF CHECK 4
Which condition is needed for the field inside a solid sphere to be exactly proportional to radius?
- Uniform density as well as spherical symmetry
- Only a spherical outer surface
- A circular satellite orbit
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SELF CHECK 5
A satellite moves from one circular orbit to a larger circular orbit around the same primary. Which endpoint statement is correct?
- Final speed and total mechanical energy are both lower
- Final speed is lower but total mechanical energy is greater
- Final speed is greater and energy is unchanged
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SELF CHECK 6
An ellipse has nearest and farthest centre distances 6.0×10⁶ m and 18.0×10⁶ m. Find its eccentricity.
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SELF CHECK 7
Two bound orbits around the same dominant primary have semimajor axes a and 4a. What is T₂/T₁?
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SELF CHECK 8
Which orbit can remain over one longitude of a rotating spherical planet in the ideal model?
- Circular, polar, with a matching period.
- Circular, equatorial, retrograde, with a matching period.
- Circular, equatorial, prograde, with a matching rotation period.
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APPLY THE CHAPTER
Take the ideas into a problem
Try a complete problem next. Commit to a method before revealing a hint, then compare your reasoning with the worked solution.
Two distances describe one ellipseIntermediate · Hints and guided reasoningEqual area is not equal angular speedAdvanced · Hints and guided reasoningSubtract the surface after finding the orbitIntermediate · Hints and guided reasoningUse the mass inside the pointIntermediate · Hints and guided reasoningCancel only the opposing componentsIntermediate · Hints and guided reasoningA wider orbit takes how much longer?Intermediate · Hints and guided reasoningEscape from a small worldIntermediate · Hints and guided reasoningEnergy of a circular orbitIntermediate · Hints and guided reasoningCOME BACK TO THE IDEA
What can you explain later?
Try these from memory at your next study session, then again a few days later. Say your reasoning aloud or sketch it before looking. Adjust the gap to what you can explain.
Next study session
Does apparent weightlessness in orbit show that gravitational force is zero?
Compare your reasoning
No. Astronaut and spacecraft share nearly the same free-fall acceleration, so the supporting contact force can vanish. A scale measures support, not gravity directly.
Revisit the explanation →A few days later: change the situation
A spacecraft moves to a higher circular orbit. Compare orbital speed and total mechanical energy in the ideal central-gravity model.
Compare your reasoning
The higher orbit has lower speed but greater total energy, meaning less negative energy. Raising gravitational potential outweighs the decrease in kinetic energy. A transfer requires a separate path and energy supply.
Revisit the explanation →Compare the physical reason, the assumptions and the conclusion. Your wording can differ. If one part is missing, revisit that section and try explaining it again after a break.
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