# How to calculate the electric energy caused by eletrostatic field stored in a region, given $V$? [closed]

I have problem in apply the rules to find the energy stored in free space here is the problem

Find the energy stored in free space for the region $$0.002<r<0.003m,\quad 0<\phi<\frac{\pi}{2},\quad 0<\theta<\frac{\pi}{2},\quad V=\frac{300\cos(\theta)}{r^2}.$$

I use the fact that $E=-\nabla V$ to get E equation and then integrate it over the volume of given region. $$W_E = \frac12\int \epsilon_0E^2dv = \frac12\epsilon_0\int E^2dv = \frac12 8.854\times10^{-12} \epsilon_0\int E^2dv$$

Then I integrate with $dv=r^2sin(\theta)\phi drd\theta d\phi$ in sphere coordinate system

so

That result

$\frac128.854\times10^{-12}\times4.34081\times10^{12}=19.2167$

I found no error in the process and the numeral calculating was done by wolframalpha but then the answer for the problem as my book says is $36.7$ so I don't know what was missed in the process or what's wrong with it.

## closed as off-topic by John Rennie, AccidentalFourierTransform, David Z♦Apr 7 '16 at 9:00

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• Can you point out which point of del is not taken care of properly, since there is no $\phi$ element in V then the derivative of V with respect to $\phi$ is obviously zero so I opt it out – aukxn Feb 2 '15 at 19:14