Question

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

• A. $\left(\rho R/6{\epsilon }_0\right)+E$
• B. $\left(\rho R/12{\epsilon }_0\right)-E$
• C. $\left(-\rho R/6{\epsilon }_0\right)+E$
• D. $\left(\rho R/12{\epsilon }_0\right)+E$

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
$=\frac{kQ}{(2R{)}^2}-E=\frac{1}{4\pi {\epsilon }_0}\frac{\rho (4/3)\pi R^3}{4R^2}-E=\frac{\rho R}{12{\epsilon }_0}-E$