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

Answer:

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  

      $k = 0.903$

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

             $E = \frac{1}{2 } * C * V ^2$

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  

      $k = 0.903$