In the book of Griffith intro to electrodynamics, on page 94, the energy of continuous charge distribution is derived in the following way:

W(total energy) = $\frac{1}{2} \int\rho V d\tau$, where $\rho$ is volume charge density, V is potential at that point and $d\tau$ is infinitesimal volume element.

Then, using $\rho = \epsilon_0 \nabla \cdot E$, one gets:

$\begin{align} W = \frac{\epsilon_0}{2} (-\int (\nabla\cdot E)V d\tau \qquad.......(1)\end{align}$

After which, he uses integration by parts to get:

$\begin{align} W = \frac{\epsilon_0}{2} (-\int (\nabla V)\cdot \,E\ d\tau\quad + \oint V\,E \cdot da ) \qquad.........(2)\end{align}$

I am unable to understand how one goes from equation 1 to 2. Please provide me detailed process rather than hints.


1 Answer 1


He use the vector formula :

$\overrightarrow{\nabla }\centerdot (V\overrightarrow{E})=V(\overrightarrow{\nabla }\centerdot \overrightarrow{E})+(\overrightarrow{\nabla }V)\centerdot \overrightarrow{E}$

And then use the Green formula for $\overrightarrow{\nabla }\centerdot (V\overrightarrow{E})$


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.