Question

If some charge is given to a solid metallic sphere, the field inside remains zero and by Gauss’s law all the charge resides on the surface. Now, suppose that Coulomb’s force between two charges varies as $1/r^3$. Then, for a charged solid metallic sphere

• A. field inside will be zero and charge density inside will be zero
• B. field inside will not be zero and charge density inside will not be zero
• C. field inside will not be zero and charge density inside will be zero
• D. field inside will be zero and charge density inside will not be zero

Answer 1

Answer: (D)

As Coulomb’s force
$F\alpha \frac{1}{r^3}$
$\Rightarrow$ Electric field, $E\alpha \frac{1}{r^3}$
$\Rightarrow E=\frac{kq}{r^3}$

Now, for Gaussian surface $(r<R)$.
Electric flux linked with surface is
$\varphi =\int \frac{kq}{r^3}\cdot 2\pi rdr$
$=2\pi kq\int \frac{dr}{r^2}=2\pi kq\left(-\frac{1}{r}\right)$
As flux is non-zero, charge density is also non-zero.
Also, by symmetry of charge distribution electric field is zero.