Question

A thin uniformly charged ring of radius $1\text{m}$, charge $+10\text{C}$ and a semi-infinite charged wire of $\lambda =+10\text{nC/m}$, oriented along the axis of ring with one of its ends coinciding with the centre of the ring, form a system. Find the force of interaction between the ring and wire.

• A. $600\text{N}$
• B. $700\text{N}$
• C. $800\text{N}$
• D. $900\text{N}$

Answer 1

The correct option is D $900\text{N}$

Given:
Charge on ring, $Q=+10\text{C};R=1\text{m}$
Linear charge density of wire, $\lambda =+10\text{nC/m}$


Considering two symmetrically infinitesimal elements $dl$ on the ring and that segment has a charge $dQ$ and acts as a point charge.

All the radial force components on charge elements $\left(dQ\right)$ get cancelled due to symmetry.

So, the parallel component of force $\left(dF\right)$ interaction between $dQ$ and semi-infinite charged wire will be,

$d{F}_{||}=\frac{dQk\lambda }{R}$

doing integration of parallel force components

$F_{||}=\int \frac{dQk\lambda }{R}$

$\Rightarrow F_{||}=\frac{k\lambda }{R}\int dQ$

$\Rightarrow F_{||}=\frac{k\lambda Q}{R}$

$\Rightarrow F_{||}=\frac{9\times {10}^9\times 10\times {10}^{-9}\times 10}{1}$

$\Rightarrow F_{||}=900\text{N}$

Hence, option $\left(b\right)$ is the correct answer.