Question

117. A charge Q is distributed over a line of length L. Another
point charge q is placed at a distance r from the centre of the
line distribution. Then the force experienced by qualified is:​

Answer 1

Given that,

Charge = Q

Length = l

Another point charge = q

Let us consider a small element dx of the line charge at a distance x from the point charge q.

The electric field at q to small element dx is

$dE=\dfrac{\lambda dx}{4\pi\epsilon_{0}x^2}$

We know that,

The linear charge density is

$\lambda=\dfrac{Q}{l}$

So, the electric field at q is

$dE=\dfrac{Q dx}{4\pi\epsilon_{0}lx^2}$

The force on q due to small element is

$dF=qdE$

Put the value of dE

$dF=\dfrac{qQ dx}{4\pi\epsilon_{0}lx^2}$

On integration

$\int_{0}^{F}{dF}=\dfrac{qQ}{4\pi\epsilon_{0}l}\int_{r}^{r+l}{\dfrac{dx}{x^2}}$

$F=\dfrac{qQ}{4\pi\epsilon_{0}l}(\dfrac{1}{r}-\dfrac{1}{r+l})$

$F=\dfrac{qQ}{4\pi\epsilon_{0}r(r+l)}$

Hence, The force experienced by qualified is $\dfrac{qQ}{4\pi\epsilon_{0}r(r+l)}$