• question_answer An electric field is uniform in the positive X-direction for positive x and uniform with the same magnitude in the negative X-direction for negative x. It is given that $E=200\,\hat{i}\,N{{C}^{-1}}$ for x > 0 $E=-\,200\,\hat{i}\,N{{C}^{-1}}$ for x < 0 A right circular cylinder of length 20 cm and radius 5 cm has its centre at the origin and its axis along the X-axis, so that one face is at $x=+10\text{ }cm$ while other is at $x=-10\text{ }cm.$ (i) What is the net outward flux through each flat face? (ii) What is the net outward flux through the cylinder? (iii) What is the net charge inside the cylinder?

According to the question, a right circular cylinder is shown below: Circular cylinder (i) On the left face $\Delta S=-\Delta \hat{i}=\pi {{(0.05)}^{2}}$  $[\because \Delta S=\pi {{r}^{2}}=\pi {{(0.05)}^{2}}]$ $\therefore$ The outward flux through left face, ${{\phi }_{1}}=E\cdot \Delta S=-(200\hat{i})\cdot [-\pi {{(0.05)}^{2}}\hat{i}]$ $\Rightarrow$   ${{\phi }_{1}}=200\times \pi \times {{(0.05)}^{2}}$     $[\because \hat{i}\cdot \hat{i}=1]$ $\Rightarrow$   ${{\phi }_{1}}=1.57N\text{-}{{m}^{2}}{{C}^{-1}}$ Similarly outward flux through right face, ${{\phi }_{2}}=E\cdot \Delta S=(200\hat{i})\cdot [-\pi {{(0.05)}^{2}}\hat{i}]$ $\Rightarrow$   $\phi =1.57N-{{m}^{2}}{{C}^{-1}}$ Flux through each flat surface is 1.57 $N-{{m}^{2}}{{C}^{-1}}.$ (ii) Again at every point on the curved side of the cylinder $E\bot \Delta S.$ $\therefore$ Angle between electric field and surface area is $90{}^\circ .$ $\therefore$ Flux through curved surfaces ${{\phi }_{3}}=E\cdot \Delta S=E\Delta S\cos \,\,90{}^\circ =0$ $[\because \cos 90{}^\circ =0]$ Net outward flux through the cylinder, ${{\phi }_{net}}={{\phi }_{1}}+{{\phi }_{2}}+{{\phi }_{3}}=1.57+1.57+0$ $\Rightarrow \,\,{{\phi }_{net}}=3.14\,\,N\text{-}{{m}^{2}}{{C}^{-1}}$ (iii) By Gauss's theorem, net charge enclosed by cylinder,             $q={{\varepsilon }_{0}}\,\,{{\phi }_{net}}=8.85\times {{10}^{-12}}\times 3.14$ $\Rightarrow q=2.78\times {{10}^{-11}}C$