# 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.

Course: [Fields, circuits and thermal physics](https://studybrix.com/en/courses/fields-and-thermal-physics/index.md)
Level: university
Prerequisites: Exponential functions, circuits and energy.

## 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.

## Practice questions

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

Answer: 9 J

Explanation: U=CV²/2=2×9/2.

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

Answer: 2 s

Explanation: τ=RC=4×0.5.

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

A. Zero
B. 1/e
C. e
D. 1 exactly

Answer: 1/e

Explanation: The exponent is −1 at t=RC.

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

Answer: 2 C

Explanation: Q=CV.

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

Sample answer / marking guidance: At fixed C the energy is CV²/2, so replacing V with 2V multiplies V² by four.

Explanation: Credit fixed capacitance and the squared dependence.

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

A. Into thermal energy in the resistor
B. It disappears
C. It creates net charge
D. It is all retained in the electric field

Answer: Into thermal energy in the resistor

Explanation: The resistor dissipates energy while the capacitor voltage falls.

### 7. For a 10 V ideal discharge, find voltage at t=0.

Answer: 10 V

Explanation: exp(0)=1, matching the initial condition.

### 8. Describe a check that can catch a wrong sign in a discharge exponent.

Sample answer / marking guidance: For positive R,C and elapsed time, the magnitude must decrease toward zero. A positive exponent would instead grow, contradicting the passive discharge model.

Explanation: Credit limiting behaviour and positive component values.

## Further reading

- https://ocw.mit.edu/courses/8-09-classical-mechanics-iii-fall-2014/
- https://openstax.org/details/books/university-physics-volume-3

Original AI-assisted educational draft; not independently teacher-reviewed. References support further study, not copied exercises, affiliation or curriculum accreditation.