Question

The electric field in a region is given by E= E(nought)*x/l . Find charge contained inside the cubical volume bounded by surfaces x=0, x=a, y=0, y=a, z=0, z=a. Take E(nought) =5*10^3 N/C , l=2 cm and a=1 cm.

Answer 1

According to gauss theorem
$$\begin{gathered} \oint \overrightarrow{E}.d\overrightarrow{s}=\frac{q}{{\epsilon }_0} \\ Here\overrightarrow{E}=E_0\left(\frac{x}{l}\right) \\ \oint {\overrightarrow{E}}_0\left(\frac{x}{l}\right).d\overrightarrow{x}=\frac{q}{{\epsilon }_0} \\ \frac{E_0}{l}{\left(\frac{x^2}{2}\right)}_0^a=\frac{q}{{\epsilon }_0} \\ \frac{E_0}{l}\left(\frac{a^2}{2}\right)=\frac{q}{{\epsilon }_0} \\ q=\frac{E_0{a}^2}{2l}\epsilon \end{gathered}$$₀ Putting values q=E04×8.85×10^-12=2.21×10^-12E₀