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
$\phi=\int \frac{k q}{r^{3}} \cdot 2 \pi r d r$
$=2 \pi k q \int \frac{d r}{r^{2}}=2 \pi k q\left(-\frac{1}{r}\right)$
As flux is non-zero, charge density is also non-zero.
Also, by symmetry of charge distribution electric field is zero.