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Physics · University

Capacitors, RC transients and field energy

For a linear capacitor Q=CV, with stored energy CV²/2. Potential difference and capacitance must refer to the same device.

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Prerequisites

Exponential functions, circuits and energy.

Study notes

Capacitance

For a linear capacitor Q=CV, with stored energy CV²/2. Potential difference and capacitance must refer to the same device.

Time constant

An ideal RC discharge follows V(t)=V0 exp(−t/RC). The time constant is RC; discharge is asymptotic rather than abruptly complete.

Energy balance

Current through a resistor converts stored electrical energy into thermal energy. A transient circuit must respect both charge and energy accounting.

Limits

Ideal elements omit leakage, parasitic inductance and voltage-dependent effects. Numerical exercises state the idealization rather than promising device behaviour.

Worked examples

A capacitor of 2 F has voltage 3 V. Find its stored energy.

U=CV²/2=2×9/2.

9 J

For R=4 ohms and C=0.5 F, calculate the RC time constant.

τ=RC=4×0.5.

2 s

Check your understanding

  1. After one time constant during an ideal discharge, the voltage is what fraction of its initial value?

    1. Zero
    2. 1/e
    3. e
    4. 1 exactly

  2. An ideal capacitor has C=0.2 F and V=10 V. Find stored charge.

  3. Why does doubling voltage quadruple a linear capacitor's stored energy?

  4. Where does the initial stored energy go in an isolated ideal RC discharge?

    1. Into thermal energy in the resistor
    2. It disappears
    3. It creates net charge
    4. It is all retained in the electric field

Answers and marking guidance

Answers and marking notes
  1. 1/e

    The exponent is −1 at t=RC.

  2. 2 C

    Q=CV.

  3. At fixed C the energy is CV²/2, so replacing V with 2V multiplies V² by four.

    Credit fixed capacitance and the squared dependence.

  4. Into thermal energy in the resistor

    The resistor dissipates energy while the capacitor voltage falls.

Further reading and review references

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