Question

The force between two charges situated in air is F. The force between the same charges if distance between them is reduced to half and they are situated in a medium having dielectric constant $4$ is:

Answer 1

In the given question, forces between the charges in the air are given. Now we have changed the medium in which charges have been put, permittivity of medium is equal to the product of the dielectric constant and the permittivity of air. The distance between the charges is reduced to half; the force between two charges in the medium is equal to the force between two same charges present in the air.

Complete step-by-step solution:
Given: the force between two charges situated in air is F.
The formula for force between two charges in air is given by:
$F= \dfrac{1}{4 \pi \epsilon_{o}} \dfrac{qQ}{r^{2}}$
Where, q and Q are the charges.
r is the distance between two charges.
$\epsilon_{o}$ is the permittivity in air.
But when we replace air by some other medium, then permittivity of the medium comes into the formula.
In the given question, distance between the charges is reduced to half, and charges are situated in a medium of dielectric constant $4$.
The formula for force between two charges in any medium is given by:
$F’= \dfrac{1}{4 \pi \epsilon} \dfrac{qQ}{r’^{2}}$
$r’= \dfrac{r}{2}$ and $\epsilon = K \epsilon_{o}$
K is the dielectric constant.
$\epsilon = 4 \epsilon_{o}$
Now, force becomes
$F’= \dfrac{1}{4 \pi \left( 4 \epsilon_{o} \right)} \dfrac{qQ}{\dfrac{r^{2}}{4}}$
$\implies F’= \dfrac{1}{4 \pi \epsilon_{o}} \dfrac{qQ}{r^{2}}$
Here, $F’ = F$.
Hence, the force between the charges if distance between them is reduced to half and they are situated in a medium having dielectric constant $4$ is the same as the force between two same charges situated in air.

Note: A dielectric is a property of a material with poor electrical conductivity but acquires the ability to collect an electrical charge. Thus, presenting only displacement current, making it ideal for building a capacitor; to stock and refund electrical energy. The dielectric constant is determined by the ratio of the permittivity of the material to the permittivity of the vacuum. It expresses the degree to which a material can contain an electric flux.