A spoon can make your reflection look upright on one side and inverted on the other at suitable distances. Curvature redirects the reflected rays. We will use a declared sign convention so that the drawing and the algebra tell the same story.
The paraxial reflection law then explains real and virtual images, the half-radius focus, plane-mirror limits and image-motion sensitivity.
What you will learn to do
- Construct reflected rays using local spherical normals.
- Assign mirror coordinates without switching sign conventions.
- Explain why f = R/2 is a paraxial result.
- Find real and virtual images with signed magnification.
- Recover plane-mirror imaging as a zero-curvature limit.
- Distinguish transverse magnification from longitudinal image motion.
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: Spherical mirrors: images, signs and moving objects Learning goals: - Construct reflected rays using local spherical normals. - Assign mirror coordinates without switching sign conventions. - Explain why f = R/2 is a paraxial result. - Find real and virtual images with signed magnification. - Recover plane-mirror imaging as a zero-curvature limit. - Distinguish transverse magnification from longitudinal image motion. 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.
Does the law of reflection use the mirror’s overall axis or the local surface normal?
Compare your reasoning
It uses the local normal at the reflection point. On a curved mirror that normal changes with position, which is why a bundle of parallel rays can converge or diverge.
Revisit the explanation →01 / Spherical mirrors: images, signs and moving objects
A curved mirror changes the direction of the normal
A spherical mirror is a reflecting part of a sphere. At each surface point, the normal is the radius through that point. Reflection makes equal angles with this local normal, so rays striking different parts of the mirror can be redirected toward or away from one another.
A concave mirror presents the inside of the sphere to an incident beam and can converge rays that arrive parallel to its axis. A convex mirror presents the outside and makes those rays diverge as if they came from a point behind the mirror.
These statements concern a paraxial bundle: rays close to the axis and making small angles with it. A spherical mirror generally has spherical aberration for wide bundles, so rays far from the axis do not share exactly the same focus.
A ray directed along a radius toward the centre of curvature strikes normally and retraces its path. A ray parallel to the axis reflects through the paraxial focus for a concave mirror, or appears to come from the focus for a convex mirror. These two constructions locate images.
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: Spherical mirrors: images, signs and moving objects Focus: A curved mirror changes the direction of the normal Learning goals: - Construct reflected rays using local spherical normals. - Assign mirror coordinates without switching sign conventions. - Explain why f = R/2 is a paraxial result. - Find real and virtual images with signed magnification. - Recover plane-mirror imaging as a zero-curvature limit. - Distinguish transverse magnification from longitudinal image motion. 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.
02 / Spherical mirrors: images, signs and moving objects
Keep the original incident-light axis after reflection
Choose the mirror pole as x = 0 and the positive x direction along the original incident light, from left to right. Keep that axis fixed even though reflected rays return toward the left. A real object in front of the mirror then has u < 0.
A real reflected image forms in front of the mirror, so it has v < 0 in this convention. A virtual image behind the mirror has v > 0. A concave mirror facing the object has its centre and focus on the left, so R < 0 and f < 0; a convex mirror has R > 0 and f > 0.
This differs from the thin-lens signs used in the previous chapter: a converging lens had f > 0 because its outgoing real focus was on the right. The physics has not contradicted itself. Reflection reverses the direction of the outgoing beam while our axis remains fixed.
Another common convention assigns positive distances to real objects and real images and gives a concave mirror positive focal length. That is also valid with its corresponding definitions. Use one complete convention rather than combining signs from two systems.
| Quantity | Concave mirror facing the object | Convex mirror facing the object |
|---|---|---|
| Real object u | Negative | Negative |
| Radius R and focal length f | Negative | Positive |
| Real reflected image v | Negative, when it forms | Not formed for an ordinary real object |
| Virtual reflected image v | Positive, for an object inside the focal distance | Positive for an ordinary real object |
Paraxial spherical-mirror relation using one fixed Cartesian axis along the original incident light. All axial quantities are signed.
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: Spherical mirrors: images, signs and moving objects Focus: Keep the original incident-light axis after reflection Learning goals: - Construct reflected rays using local spherical normals. - Assign mirror coordinates without switching sign conventions. - Explain why f = R/2 is a paraxial result. - Find real and virtual images with signed magnification. - Recover plane-mirror imaging as a zero-curvature limit. - Distinguish transverse magnification from longitudinal image motion. Equations in this section (LaTeX): \frac1v+\frac1u=\frac1f=\frac2R,\qquad m=-\frac vu Meaning: Paraxial spherical-mirror relation using one fixed Cartesian axis along the original incident light. All axial quantities are signed. 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 / Spherical mirrors: images, signs and moving objects
The focal distance is half the radius in the paraxial limit
Consider a parallel incident ray striking a concave mirror at a small height h. The surface normal is tilted by a small angle α relative to the axis, with α approximately h/|R|. Reflection turns the outgoing ray toward the axis by approximately 2α.
The axial distance to its intersection is therefore approximately h/(2α) = |R|/2. Including the Cartesian signs gives f = R/2. The factor two comes from reflection doubling the small normal tilt.
For finite object distance, the same small-angle geometry gives the mirror relation 1/v + 1/u = 2/R. The sum of inverse distances appears because the ray approaches and then returns on the incident side.
The focal relation is approximate for a spherical mirror outside the paraxial regime. A parabolic reflecting surface has a different geometrical property for an on-axis parallel beam. Do not apply a spherical small-angle derivation to all mirror shapes and ray angles without checking its assumptions.
WORKED EXAMPLE
Assign the sign to a measured curvature radius
A concave spherical mirror faces an object on its left. Its radius magnitude is 40 cm and incident light travels to the right. Find the signed radius and focal length.
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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: Spherical mirrors: images, signs and moving objects Focus: The focal distance is half the radius in the paraxial limit Learning goals: - Construct reflected rays using local spherical normals. - Assign mirror coordinates without switching sign conventions. - Explain why f = R/2 is a paraxial result. - Find real and virtual images with signed magnification. - Recover plane-mirror imaging as a zero-curvature limit. - Distinguish transverse magnification from longitudinal image motion. 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.
04 / Spherical mirrors: images, signs and moving objects
A concave mirror can form a real inverted image
Place a real object farther from a concave mirror than its focal distance. Reflected paraxial rays can converge in front of the mirror, so the image is real. The negative magnification shows that it is inverted relative to the object.
If the object is beyond the centre of curvature, the image lies between the focus and centre and is reduced. At the centre, object and image distances are equal and the image has equal size with inversion. Between centre and focus, the real image lies beyond the centre and is enlarged.
These location rules are useful checks, but they follow from the equation and ray geometry rather than replacing them. Draw one parallel ray and one radius-directed ray to test a proposed answer.
At the focal point the reflected rays are parallel, so no finite image plane exists. Moving a screen farther and farther away is the limiting construction; assigning a finite magnification at that exact position would be incorrect.
WORKED EXAMPLE
Find position and orientation with a single sign convention
An upright object is 60 cm in front of a concave mirror with f = −20 cm. Find v and m.
MEMBERSThe approach, step-by-step working and check are for members.Unlock →
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: Spherical mirrors: images, signs and moving objects Focus: A concave mirror can form a real inverted image Learning goals: - Construct reflected rays using local spherical normals. - Assign mirror coordinates without switching sign conventions. - Explain why f = R/2 is a paraxial result. - Find real and virtual images with signed magnification. - Recover plane-mirror imaging as a zero-curvature limit. - Distinguish transverse magnification from longitudinal image motion. 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.
05 / Spherical mirrors: images, signs and moving objects
Backward extensions explain both magnifying and convex mirrors
A real object inside a concave mirror’s focal distance produces reflected rays that diverge. Their backward extensions intersect behind the mirror, giving a virtual, upright, enlarged image. This is the geometry used when a concave mirror is viewed at close range.
A convex mirror with an ordinary real object also produces a virtual upright image, but it is reduced and lies between the pole and the virtual focus behind the mirror. The reflected bundle diverges more than a parallel bundle.
The formulas distinguish the two cases through the signed focal length. Both can give v > 0 and m > 0, but the magnification magnitude differs. A positive magnification means upright, not automatically enlarged.
A screen behind the reflecting surface cannot capture either virtual image from these outgoing rays. Seeing the image with an eye is still possible because the eye’s lens forms its own real retinal image from the received light.
WORKED EXAMPLE
A close object in a concave mirror
A concave mirror has f = −10 cm. A real object is at u = −6 cm. Find the image coordinate and magnification.
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WORKED EXAMPLE
A convex mirror gives a reduced upright image
A convex mirror has f = +12 cm and a real object at u = −24 cm. Find v and m.
MEMBERSThe approach, step-by-step working and check are for members.Unlock →
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: Spherical mirrors: images, signs and moving objects Focus: Backward extensions explain both magnifying and convex mirrors Learning goals: - Construct reflected rays using local spherical normals. - Assign mirror coordinates without switching sign conventions. - Explain why f = R/2 is a paraxial result. - Find real and virtual images with signed magnification. - Recover plane-mirror imaging as a zero-curvature limit. - Distinguish transverse magnification from longitudinal image motion. 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 / Spherical mirrors: images, signs and moving objects
A plane mirror provides a useful limiting check
As the radius of curvature grows without bound, the mirror becomes locally plane and 1/f tends to zero. The mirror equation becomes 1/v + 1/u = 0, so v = −u.
For a real object at negative u, the image is at equal positive v behind the plane. Its magnification is −v/u = +1, so it is upright and the same transverse size.
The equal-distance construction follows from the reflection law and symmetry. It also shows why a mirror image’s apparent position is not a physical point through which the reflected rays pass.
Changing the mirror’s position changes the origin as well as the image coordinate relative to the mirror. A moving-mirror problem therefore needs positions expressed in a common laboratory frame before differentiating.
WORKED EXAMPLE
Convert a relative mirror image into a laboratory position
A plane mirror is at laboratory x = 10 cm. An object is at x = 2 cm on its incident side. Find the image’s laboratory position.
MEMBERSThe approach, step-by-step working and check are for members.Unlock →
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: Spherical mirrors: images, signs and moving objects Focus: A plane mirror provides a useful limiting check Learning goals: - Construct reflected rays using local spherical normals. - Assign mirror coordinates without switching sign conventions. - Explain why f = R/2 is a paraxial result. - Find real and virtual images with signed magnification. - Recover plane-mirror imaging as a zero-curvature limit. - Distinguish transverse magnification from longitudinal image motion. 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 / Spherical mirrors: images, signs and moving objects
Image motion can be faster than object motion
For a fixed spherical mirror, differentiate 1/v + 1/u = 1/f with f constant. This gives dv/du = −v²/u² = −m². A small object displacement and corresponding image displacement therefore have opposite signed directions in the fixed axis.
The magnitude ratio of their axial velocities is m², not |m|. Lateral magnification describes transverse heights; longitudinal response describes axial shifts. They are related but not identical.
Near a focal configuration, the image can move a large distance when the object moves only a little. This sensitivity explains why sharp focusing becomes difficult near certain geometries and why a first-order displacement approximation needs a sufficiently small change.
If the mirror moves, f may be constant relative to the mirror but u and v are measured from a moving origin. The derivative above is for a fixed mirror and cannot be used directly as a laboratory-velocity relation without the coordinate transformation.
WORKED EXAMPLE
Use a local derivative for a small object displacement
For a fixed concave mirror, an object is at u = −30 cm and its image at v = −60 cm. Estimate the image displacement for a small object-coordinate increase Δu = +0.10 cm.
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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: Spherical mirrors: images, signs and moving objects Focus: Image motion can be faster than object motion Learning goals: - Construct reflected rays using local spherical normals. - Assign mirror coordinates without switching sign conventions. - Explain why f = R/2 is a paraxial result. - Find real and virtual images with signed magnification. - Recover plane-mirror imaging as a zero-curvature limit. - Distinguish transverse magnification from longitudinal image motion. 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.
BRING THE IDEAS TOGETHER
What should stay with you
- Mirror coordinates remain on the original incident-light axis after reflection.
- In this Cartesian convention a concave mirror facing a real object has negative R and f.
- The mirror equation is 1/v + 1/u = 1/f and m = −v/u.
- A real image is formed by reflected rays; a virtual image by their backward extensions.
- The plane limit gives v = −u and unit upright magnification.
- For a fixed mirror, local axial sensitivity is dv/du = −m².
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
A concave mirror has signed radius R = −30 cm. What is its paraxial focal length?
Answer in cm.
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SELF CHECK 2
A concave mirror has f = −15 cm and object coordinate u = −30 cm. What is v?
Answer in cm.
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SELF CHECK 3
A plane mirror is at laboratory x = 12 cm and an object at x = 5 cm. Where is its image in the laboratory frame?
Answer in cm.
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SELF CHECK 4
In this fixed Cartesian mirror convention, a real reflected image in front of the mirror has:
- v < 0
- v > 0
- No signed position
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SELF CHECK 5
For a convex mirror and an ordinary real object, the image is:
- Real, inverted and enlarged
- Virtual, upright and reduced
- Always at infinity
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SELF CHECK 6
For a fixed mirror with |m| = 3, the magnitude of the local axial image displacement per unit object displacement is:
- 3
- 6
- 9
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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.
Locate a real mirror imageIntermediate · Hints and guided reasoningRead the sign of a virtual imageIntermediate · 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
Why can a convex mirror show an upright virtual image without rays meeting behind the mirror?
Compare your reasoning
The reflected rays diverge in front, but their backward extensions meet behind the mirror. The eye interprets their directions as if they came from that apparent location.
Revisit the explanation →A few days later: change the situation
As a spherical mirror’s curvature becomes very small, what image behaviour should your formula approach?
Compare your reasoning
It should approach a plane mirror: image and object lie at equal distances on opposite sides in the usual geometry, with an upright image of equal size. A limiting case checks the sign convention.
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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