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/6ε_{0} + E

(b) ρR/12ε_{0} – E

(c) -ρR/6ε_{0} + E

(d) ρR/12ε_{0} + 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 $