RC transients | PhysicsRank

Close a switch and a capacitor does not usually jump straight to its final voltage. The resistor limits the charging current, and the changing capacitor voltage changes that current in return. Following the feedback is more useful than memorizing one exponential.

We will derive the response, distinguish voltage and energy decay, find the effective resistance of a network and close the source–capacitor–resistor energy balance.

What you will learn to do

Use capacitor-voltage continuity at a finite-current switch.
Derive charging and discharging equations from a loop balance.
Calculate threshold times from the remaining voltage fraction.
Distinguish voltage and stored-energy decay rates.
Find initial value, final value and effective RC for a network.
Carry the actual capacitor state through multiple stages.
Account for source energy, field storage and resistor loss.

Useful before this chapterPotential, capacitors and dipoles →Current, resistance and circuit constraints →Exponential change, logarithms and controlled approximations →

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: RC switching: initial states, time constants and energy
Learning goals:
- Use capacitor-voltage continuity at a finite-current switch.
- Derive charging and discharging equations from a loop balance.
- Calculate threshold times from the remaining voltage fraction.
- Distinguish voltage and stored-energy decay rates.
- Find initial value, final value and effective RC for a network.
- Carry the actual capacitor state through multiple stages.
- Account for source energy, field storage and resistor loss.

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.

Which capacitor quantity stays continuous at a switching instant if current remains finite?

Compare your reasoning

Voltage stays continuous for a finite constant capacitance, because changing its charge by a finite amount takes a finite current–time integral. The current itself can change abruptly in an ideal RC model.

Revisit the explanation →

01 / RC switching: initial states, time constants and energy

A capacitor remembers its voltage when a switch changes the circuit

A capacitor’s voltage is related to stored charge by Q = CV. Changing that voltage requires charge to move. For a fixed capacitance, current into the positively labelled plate is i_C = C dv_C/dt.

If the current remains finite, capacitor voltage cannot jump instantaneously. Immediately after a switch operation, v_C(0⁺) equals v_C(0⁻). The surrounding resistor currents can change abruptly in the ideal lumped model because their values respond to the new connections and the existing capacitor voltage.

The rule that a capacitor initially behaves like a short circuit applies only to a capacitor with zero initial voltage in an appropriate circuit. A precharged capacitor initially acts as its existing voltage constraint. Replacing every capacitor by a wire at t = 0 discards information that can determine the current direction.

In a long-time DC steady state, the capacitor current is zero and its branch can be treated as open for finding the final voltage. Other resistor branches may still carry current. Open capacitor branch does not mean zero current everywhere in the circuit.

i_C=C\frac{dv_C}{dt},\qquad v_C(0^+)=v_C(0^-)\ \text{for finite current}

Current is positive into the plate labelled positive for v_C. A directly imposed ideal voltage jump would require an impulse current and lies outside the finite-R switching model.

STOP & THINK

A capacitor already has 4 V across it when a switch connects it through a finite resistor to a new source. What is its voltage immediately afterward?

Zero because switching resets capacitors
4 V with the same polarity
The new source voltage immediately

MEMBERSAnswer checking, feedback and the explanation 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: RC switching: initial states, time constants and energy
Focus: A capacitor remembers its voltage when a switch changes the circuit
Learning goals:
- Use capacitor-voltage continuity at a finite-current switch.
- Derive charging and discharging equations from a loop balance.
- Calculate threshold times from the remaining voltage fraction.
- Distinguish voltage and stored-energy decay rates.
- Find initial value, final value and effective RC for a network.
- Carry the actual capacitor state through multiple stages.
- Account for source energy, field storage and resistor loss.
Equations in this section (LaTeX):
i_C=C\frac{dv_C}{dt},\qquad v_C(0^+)=v_C(0^-)\ \text{for finite current}
Meaning: Current is positive into the plate labelled positive for v_C. A directly imposed ideal voltage jump would require an impulse current and lies outside the finite-R switching 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.

Back to chapter contents ↑

02 / RC switching: initial states, time constants and energy

The growing capacitor voltage reduces the resistor current

Consider an initially uncharged capacitor connected in series with resistance R to an ideal DC source V_s. The loop equation is V_s − iR − v_C = 0. Substituting i = C dv_C/dt gives RC dv_C/dt + v_C = V_s.

Initially v_C = 0, so all the source voltage appears across the resistor and i(0) = V_s/R. As charge accumulates, v_C rises and the resistor voltage V_s − v_C falls. The current therefore decreases while the capacitor continues charging.

Solving the first-order equation gives v_C = V_s(1 − e^(−t/RC)) and i = (V_s/R)e^(−t/RC). The capacitor voltage approaches V_s asymptotically in the ideal model; it does not suddenly finish charging at one time constant.

The time constant τ = RC has units seconds because Ω·F = (V/A)(C/V) = C/A. Larger resistance limits charge flow more strongly, and larger capacitance requires more charge for a given voltage change. Both increase the response time.

v_C(t)=V_s(1-e^{-t/\tau}),\qquad i(t)=\frac{V_s}{R}e^{-t/\tau},\qquad\tau=RC

Initially uncharged capacitor, ideal fixed DC source, constant positive R and C, negligible inductance and leakage.

Follow voltage growth and current decay

Inspect a plotted point: t/τ = 0

v_C / V_s: 0
i / i₀: 1

The curves are normalized to final capacitor voltage and initial current. At one time constant, the voltage is about 0.632 of final and the current about 0.368 of initial. Lines connect the plotted samples.

WORKED EXAMPLE

Find the state after one time constant

An initially uncharged 2.0 μF capacitor charges from a 10 V source through 1.0 MΩ. Find the time constant, initial current and capacitor voltage after 2.0 s.

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: RC switching: initial states, time constants and energy
Focus: The growing capacitor voltage reduces the resistor current
Learning goals:
- Use capacitor-voltage continuity at a finite-current switch.
- Derive charging and discharging equations from a loop balance.
- Calculate threshold times from the remaining voltage fraction.
- Distinguish voltage and stored-energy decay rates.
- Find initial value, final value and effective RC for a network.
- Carry the actual capacitor state through multiple stages.
- Account for source energy, field storage and resistor loss.
Equations in this section (LaTeX):
v_C(t)=V_s(1-e^{-t/\tau}),\qquad i(t)=\frac{V_s}{R}e^{-t/\tau},\qquad\tau=RC
Meaning: Initially uncharged capacitor, ideal fixed DC source, constant positive R and C, negligible inductance and leakage.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Back to chapter contents ↑

03 / RC switching: initial states, time constants and energy

A target fraction determines a logarithmic waiting time

To reach a fraction f of the final voltage from zero, the unfilled fraction is 1 − f = e^(−t/τ). Taking logarithms gives t = −τ ln(1 − f). The argument is dimensionless and positive for 0 < f < 1.

Reaching 90% requires τ ln10, about 2.303τ. Reaching 99% requires τ ln100, about 4.605τ. Each extra factor of ten reduction in the remaining error costs another τ ln10, rather than the same fixed increase in percentage.

A measured charging trace can estimate τ from the decay of the voltage gap V_s − v_C. The logarithm of that gap is linear in time if the ideal model holds. Using ln v_C directly for a zero-start charging curve would fit the wrong variable.

A real measurement includes source resistance, capacitor leakage and meter loading. The measured time constant belongs to the effective circuit seen by the capacitor, which may differ from the product of the two most visually prominent component labels.

WORKED EXAMPLE

Calculate a threshold time instead of calling the capacitor fully charged

For the preceding RC circuit with τ = 2.0 s, how long does it take to reach 90% of the final voltage?

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: RC switching: initial states, time constants and energy
Focus: A target fraction determines a logarithmic waiting time
Learning goals:
- Use capacitor-voltage continuity at a finite-current switch.
- Derive charging and discharging equations from a loop balance.
- Calculate threshold times from the remaining voltage fraction.
- Distinguish voltage and stored-energy decay rates.
- Find initial value, final value and effective RC for a network.
- Carry the actual capacitor state through multiple stages.
- Account for source energy, field storage and resistor loss.
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.

Back to chapter contents ↑

04 / RC switching: initial states, time constants and energy

During discharge, voltage and charge decay while the current sign depends on your arrow

Disconnect the source and connect a capacitor initially at voltage V₀ across a resistor R. With current defined into the positive plate, the loop and capacitor relation give dv_C/dt = −v_C/(RC). Thus v_C = V₀e^(−t/τ).

The current into the positive plate is negative: i_C = −(V₀/R)e^(−t/τ), because positive-plate charge decreases. If you instead label the discharge current leaving that plate as positive, its magnitude is v_C/R. Both conventions describe the same motion of charge.

Stored energy is ½Cv_C², so it decays as e^(−2t/τ). Its decay time constant is τ/2, different from the voltage time constant. After one τ, the voltage is 1/e of its initial value but energy is 1/e² of its initial value.

The resistor dissipates the decreasing field energy as heat in the ideal discharge circuit. At each instant I²R equals −dU_C/dt. This rate check is a useful independent verification of both the exponential and its factor of two in energy.

v_C=V_0e^{-t/\tau},\qquad U_C=\frac12CV_0^2e^{-2t/\tau}

Source-free resistor discharge with the capacitor initially at V₀. Current direction must be specified separately.

WORKED EXAMPLE

Distinguish a voltage half-life from an energy half-life

A capacitor discharges through a resistor with τ = 4.0 s. Find the time for its stored energy to fall to half the initial value.

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: RC switching: initial states, time constants and energy
Focus: During discharge, voltage and charge decay while the current sign depends on your arrow
Learning goals:
- Use capacitor-voltage continuity at a finite-current switch.
- Derive charging and discharging equations from a loop balance.
- Calculate threshold times from the remaining voltage fraction.
- Distinguish voltage and stored-energy decay rates.
- Find initial value, final value and effective RC for a network.
- Carry the actual capacitor state through multiple stages.
- Account for source energy, field storage and resistor loss.
Equations in this section (LaTeX):
v_C=V_0e^{-t/\tau},\qquad U_C=\frac12CV_0^2e^{-2t/\tau}
Meaning: Source-free resistor discharge with the capacitor initially at V₀. Current direction must be specified separately.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Back to chapter contents ↑

05 / RC switching: initial states, time constants and energy

Use initial value, final value and one time constant

For a single-capacitor first-order circuit connected through a linear resistance to a constant final source condition, the voltage has the form v_C(t) = V_∞ + (V₀ − V_∞)e^(−t/τ). It starts at V₀ and approaches V_∞, whether that means charging, discharging or reversing polarity.

The initial current depends on the voltage difference V_∞ − V₀, not on V_∞ alone. If the capacitor begins above the source voltage, current initially flows back toward the source through the resistor. An ideal source can absorb energy in this model.

For a linear network of resistors and independent sources, find V_∞ as the open-circuit voltage at the capacitor terminals in the final DC state. Find the resistance seen from those terminals with independent ideal voltage sources replaced by shorts and independent ideal current sources by opens. Then τ = R_thC.

Dependent sources cannot simply be turned off by that rule; a test source may be needed to find the driving-point resistance. Multiple independent capacitors or inductors can introduce more than one time constant. The compact single-exponential expression requires the first-order model.

v_C(t)=V_\infty+(V_0-V_\infty)e^{-t/(R_{\mathrm{th}}C)}

One independent capacitor state and a linear time-invariant resistive source network under the stated switch position.

WORKED EXAMPLE

Find the resistance actually seen by the capacitor

A 3 kΩ resistor connects an ideal 12 V source to a node. A 6 kΩ resistor connects that node to ground. A 1 μF capacitor also connects the node to ground and initially has 2 V across it. Find the final voltage, time constant and voltage after one time constant.

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: RC switching: initial states, time constants and energy
Focus: Use initial value, final value and one time constant
Learning goals:
- Use capacitor-voltage continuity at a finite-current switch.
- Derive charging and discharging equations from a loop balance.
- Calculate threshold times from the remaining voltage fraction.
- Distinguish voltage and stored-energy decay rates.
- Find initial value, final value and effective RC for a network.
- Carry the actual capacitor state through multiple stages.
- Account for source energy, field storage and resistor loss.
Equations in this section (LaTeX):
v_C(t)=V_\infty+(V_0-V_\infty)e^{-t/(R_{\mathrm{th}}C)}
Meaning: One independent capacitor state and a linear time-invariant resistive source network under the stated switch position.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Back to chapter contents ↑

06 / RC switching: initial states, time constants and energy

Carry the final state of one stage into the next

For a sequence of switch positions, solve each stage with its own circuit, final target and time constant. At the switching instant, carry the capacitor voltage continuously into the next stage, but reset the local time variable for the new exponential.

The capacitor may not have reached its previous asymptotic state when switching occurs. Using the old source voltage as the next initial condition without calculating the actual stored voltage can overestimate both charge and current.

A stage table can record switch configuration, initial voltage, final target, effective resistance and duration. This separates topology from time evolution and makes sign changes easier to follow.

Check the endpoints of every stage: substitute zero local time to recover the inherited voltage, and take long time to recover the new target. Those checks are often more informative than memorizing a separate formula for every switch drawing.

WORKED EXAMPLE

Switch before charging has finished

An initially uncharged capacitor charges toward 12 V for exactly one time constant τ. It is then disconnected from the source and discharged through a resistor giving the same τ. Find the voltage after one further time constant.

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: RC switching: initial states, time constants and energy
Focus: Carry the final state of one stage into the next
Learning goals:
- Use capacitor-voltage continuity at a finite-current switch.
- Derive charging and discharging equations from a loop balance.
- Calculate threshold times from the remaining voltage fraction.
- Distinguish voltage and stored-energy decay rates.
- Find initial value, final value and effective RC for a network.
- Carry the actual capacitor state through multiple stages.
- Account for source energy, field storage and resistor loss.
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.

Back to chapter contents ↑

07 / RC switching: initial states, time constants and energy

The source, capacitor and resistor need separate energy entries

Charging an initially uncharged capacitor from a constant ideal source V_s transfers total charge CV_s. The source supplies energy V_s(CV_s) = CV_s². The final field energy is only ½CV_s²; the other half is dissipated in the resistor in the ideal RC model.

The total dissipated energy is independent of the positive resistance value, although the power history and time constant depend on R. Reducing R increases the initial current and shortens the process. The exact zero-resistance limit needs other circuit effects rather than an instantaneous dissipation-free version of the same RC solution.

For an initial capacitor voltage V₀ connected through R to a fixed source V_s, the source energy transfer is CV_s(V_s − V₀), the field-energy change is ½C(V_s² − V₀²), and their difference is ½C(V_s − V₀)² dissipated. The source transfer can be negative if it absorbs energy.

These expressions assume the transient is allowed to settle and the source remains ideal and fixed. A switched finite-duration stage should instead integrate the actual current or use the charge transferred during that stage.

WORKED EXAMPLE

Account for charging a precharged capacitor

A 4 μF capacitor initially at 8 V is connected through a positive resistor to an ideal 12 V source and allowed to settle. Find source energy supplied, field-energy increase and resistor dissipation.

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: RC switching: initial states, time constants and energy
Focus: The source, capacitor and resistor need separate energy entries
Learning goals:
- Use capacitor-voltage continuity at a finite-current switch.
- Derive charging and discharging equations from a loop balance.
- Calculate threshold times from the remaining voltage fraction.
- Distinguish voltage and stored-energy decay rates.
- Find initial value, final value and effective RC for a network.
- Carry the actual capacitor state through multiple stages.
- Account for source energy, field storage and resistor loss.
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.

Back to chapter contents ↑

BRING THE IDEAS TOGETHER

What should stay with you

A capacitor with finite current keeps its voltage continuous through switching.
The initial short-circuit shortcut applies only to an initially uncharged capacitor.
The RC time constant sets an exponential approach, not an exact finishing time.
Stored energy decays twice as fast in the exponent as voltage during discharge.
Use the resistance seen by the capacitor under the actual circuit topology.
A final open capacitor branch can coexist with nonzero currents elsewhere.
Charge continuity and complete energy accounting are separate checks.

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 5 kΩ resistance is seen by a 4 μF capacitor. What is the time constant in milliseconds?

Answer in ms.

MEMBERSAnswer checking, feedback and the explanation are for members.Unlock →

SELF CHECK 2

A capacitor discharges with τ = 3 s. What fraction of its initial voltage remains after 3 ln2 seconds?

MEMBERSAnswer checking, feedback and the explanation are for members.Unlock →

SELF CHECK 3

After one discharge time constant, the remaining energy fraction is what?

1/e
1/e²
Zero

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SELF CHECK 4

In a DC divider with a capacitor connected from its midpoint to ground, what does long-time zero capacitor current imply?

Every resistor current is zero
Only the capacitor branch has zero steady current
The capacitor voltage must be zero

MEMBERSAnswer checking, feedback and the explanation are for members.Unlock →

SELF CHECK 5

An initially uncharged capacitor is fully charged through R by a constant ideal source. What fraction of source energy is stored in the capacitor in the ideal RC model?

All of it
One half
None

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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.

A capacitor that does not start emptyIntermediate · Hints and guided reasoning Half the voltage, one quarter of the energyIntermediate · Hints and guided reasoning Read capacitance from a logarithmic graphIntermediate · Hints and guided reasoning Where does the battery energy go?Advanced · Hints and guided reasoning

COME 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

A capacitor starts at a nonzero voltage before being connected to a new DC circuit. Which three quantities organize its exponential response?

Compare your reasoning

Its initial voltage, final steady voltage and the effective time constant. The remaining difference from the final value decays exponentially; starting from zero is only one special case.

Revisit the explanation →

A few days later: change the situation

An initially uncharged capacitor charges from an ideal constant-voltage source through a resistor. Does all the source work remain stored in the capacitor?

Compare your reasoning

No. In this ideal charging process half the source work becomes capacitor energy and half becomes resistor heat. The resistance changes the timescale, while the fixed-voltage endpoint energy split stays the same.

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.

Back to chapter contents ↑