Question

In an RC circuit, what fraction of the final energy is stored in an initially uncharged capacitor after it has been charging for 3.0 time constants

Answer 1

The fraction fraction of the final energy is stored in an initially uncharged capacitor after it has been charging for 3.0 time constants is

`k = 0.903`

Step-by-step explanation:

From the question we are told that

The time constant
`\tau = 3`

The potential across the capacitor can be mathematically represented as

`V = V_o (1 - e^(- \tau))`

Where
`V_o` is the voltage of the capacitor when it is fully charged

So at
`\tau = 3`

`V = V_o (1 - e^(- 3))`

`V = 0.950213 V_o`

Generally energy stored in a capacitor is mathematically represented as

`E = (1)/(2 ) * C * V ^2`

In this equation the energy stored is directly proportional to the the square of the potential across the capacitor

Now since capacitance is constant at
`\tau = 3`

The energy stored can be evaluated at as

`V^2 = (0.950213 V_o )^2`

`V^2 = 0.903 V_o ^2`

Hence the fraction of the energy stored in an initially uncharged capacitor is

`k = 0.903`