Question

A non-conducting infinite rod is placed along the $z$-axis, the upper half of the rod (lying along $z \geq 0$ ) is chárged positively with a uniform linear charge density $+\lambda$ while the lower half $(z < 0)$ is charged negatively with a uniform linear charge density $-\lambda$. The origin is located at the junction of the positive and negative halves of the rod. A uniformly charged annular disc (surface charge density: $\sigma_{0}$ ) of inner radius $R$ and outer radius $2 R$ is placed in the $x-y$ plane with its centre at the origin. The force on the rod due to the disc is $\frac{y \sigma_{0} \lambda R}{8 \varepsilon_{0}}$. Find the value of $y$.

Answer 1

The force on the rod due to the annular disc is equal and opposite to that on the disc due to rod.
We take an annular strip of radius $r$ and width $d r$ and find the force acting on it.
$d F=\left(\sigma_{0} 2 \pi r d r\right) \frac{\lambda}{2 \pi \varepsilon_{0} r}=\frac{\lambda \sigma_{0}}{\varepsilon_{0}} d r$
Total force, $F=\frac{\sigma_{0} \lambda}{\varepsilon_{0}} \int\limits_{R}^{2 R} d r=\frac{\sigma_{0} \lambda R}{\varepsilon_{0}}$