Q. Figure shows a uniformly charged hemisphere of radius R. It has a volume charge density ρ . If the electric field at a point 2R , above its center is E, then what is the electric field at the point 2R below it center ?
(a) ρR/6ε0 + E
(b) ρR/12ε0 – E
(c) -ρR/6ε0 + E
(d) ρR/12ε0 + E
Ans: (b)
Sol: Consider a complete a sphere Lower & upper half (which is absent)
Net field at A due to Lower half & upper half is
$ \displaystyle E_L + E_U = \frac{1}{4\pi \epsilon_0} \frac{Q}{(2R)^2}$
$ \displaystyle Q = \rho \times \frac{4}{3} \pi R^3$
$ \displaystyle E_L + E_U = \frac{1}{4\pi \epsilon_0 (4 R^2)} \times \rho \times \frac{4}{3} \pi R^3 $
$ \displaystyle E_L + E_U = \frac{\rho R}{12 \epsilon_0 } $
$ \displaystyle E_U = \frac{\rho R}{12 \epsilon_0 } – E_L $
$ \displaystyle E_U = \frac{\rho R}{12 \epsilon_0 } – E $