Question

A hollow cylindrical box of length $1m$ and area of cross-section $25c{m}^2$ is placed in a three dimensional coordinate system as shown in the figure. The electric field in the region is given by $\overrightarrow{E}=50x\hat{i}$, where $E$ is in $N{C}^{-1}$ and $x$ is in metres.
Find
(i) Net flux through the cylinder.
(ii) Charge enclosed by the cylinder.

Answer 1

Area of the cross-section of the cylinder $A=25\times {10}^{-4}{m}^{-2}$
Electric field at the face nearer to the origin $E_1=20\times 0.5=10N/C$
Thus flux entering through this face ${\varphi }_1=E_{1}A=10\times 25\times {10}^{-4}=2.5\times {10}^{-2}Wb$
Distance of second face from origin = $1+0.5=1.5m$
Electric field at the face away from the origin $E_2=20\times \left(1.5\right)=30N/C$
Thus flux leaving out through this face ${\varphi }_2=E_{2}A=30\times 25\times {10}^{-4}=7.5\times {10}^{-2}Wb$
(i) : Net flux through the cylinder $\varphi =(7.5-2.5)\times {10}^{-2}=5\times {10}^{-2}Wb$
(ii) : Charge enclosed in the cylinder $Q=\varphi {ϵ}_o$
$\therefore$ $Q=5\times {10}^{-2}\times 8.85\times {10}^{-12}=4.42\times {10}^{-13}C$