Question

in a certain region of free space a non-uniform electric field which depends on x-coordinates is given by $\overrightarrow{E}=E_{0}x\hat{i}$. What is the total amount of electric potential energy contained within the cube of sides of length L as shown below?

• A. $\frac{1}{5}{\epsilon }_0{E}_0^2{L}^5$
• B. $\frac{1}{5}{\epsilon }_0{E}_0^2{L}^6$
• C. $\frac{1}{6}{\epsilon }_0{E}_0^2{L}^6$
• D. $\frac{1}{2}{\epsilon }_0{E}_0^2{L}^5$

Answer 1

Answer: (D)

$\begin{array}{c}fromthegivenfigure\\ du=\frac{1}{2}{\epsilon }_0\left(t^2\right).\left(1^{2}dx\right)\\ =\frac{1}{2}{\epsilon }_o{L}^2{E}_0^2{\int }_0^L{x}^{2}dx\\ =\frac{1}{2}{\epsilon }_0{L}^2{E}_0^2\left(\frac{L^3}{3}\right)\\ =\frac{1}{2}{\epsilon }_0{L}^5{E}_0^2.\\ Hence,theoptionDisthecorrectanswer.\end{array}$