Question

Capacitor 2 has half the capacitance and twice the potential difference as capacitor 1. What is the ratio (U_{\rm C})_1/\,(U_{\rm C})_2.

Answer 1

1/2

Step-by-step explanation:

The energy stored in a capacitor is given by

`U=(1)/(2)CV^2`

where

C is the capacitance

V is the potential difference

Calling
`C_1` the capacitance of capacitor 1 and
`V_1` its potential difference, the energy stored in capacitor 1 is

`U=(1)/(2)C_1 V_1^2`

For capacitor 2, we have:

- The capacitance is half that of capacitor 1:
`C_2 = (C_1)/(2)`

- The voltage is twice the voltage of capacitor 1:
`V_2 = 2 V_1`

so the energy stored in capacitor 2 is

`U_2 = (1)/(2)C_2 V_2^2 = (1)/(2)(C_1)/(2)(2V_1)^2 = C_1 V_1^2`

So the ratio between the two energies is

`(U_1)/(U_2)=((1)/(2)C_1 V_1^2)/(C_1 V_1^2)=(1)/(2)`