Question

The diagram shows a uniformly charged hemisphere of radius $R .$ It has volume charge density $\rho .$ If the electric field at a point $2 R$ distance above its centre is $E$, then what is the electric field at the point which is $2 R$ below its centre?

• A. $\left(\rho R / 6 \varepsilon_{0}\right)+E$
• B. $\left(\rho R / 12 \varepsilon_{0}\right)-E$
• C. $\left(-\rho R / 6 \varepsilon_{0}\right)+E$
• D. $\left(\rho R / 12 \varepsilon_{0}\right)+E$

Answer 1

Answer: (B)

Let us complete the sphere. Electric field at $A$ due to lower part $=$ electric field at $B$ due to upper part $=E$ (given)
Electric field at $B$ due to lower part = electric field due to full sphere - electric field due to upper part
$=\frac{k Q}{(2 R)^{2}}-E=\frac{1}{4 \pi \varepsilon_{0}} \frac{\rho(4 / 3) \pi R^{3}}{4 R^{2}}-E=\frac{\rho R}{12 \varepsilon_{0}}-E$