Question

Two identical rings of radii $0.1m$ are placed co-axially at a distance $0.5m$ apart. The charges on the rings are $2\mu C$ and $4\mu C$ respectively. The work done in transferring $5\mu C$ charge from centre of one ring to other is

Answer 1

R.E.F image

The charge $Q_o=5\mu c$ is transferred from $A\to B,d=0.5m$

Potential due to a charged ring at a distance x from

In center In given by :-

$v=\frac{KQ}{\sqrt{r^2+x^2}}$ where $k=\frac{1}{4\pi E_o},r=0.1m$

Potential at center A,

$V_A=\frac{k{q}_1}{r}+\frac{k{q}_2}{\sqrt{r^2+d^2}}$ [potential due to both rings]

$=q\times {10}^q(\frac{2\times {10}^{-6}}{0.1}+\frac{4\times {10}^{-6}}{\sqrt{{0.1}^2+{0.5}^2}})=2.5\times {10}^{5}V$

Similarity at B,

$V_B=\frac{k{q}_2}{r}+\frac{k{q}_2}{\sqrt{r^2+d^2}}$

$=9\times {10}^9(\frac{4\times {10}^{-6}}{0.1}+\frac{2\times {10}^{-6}}{\sqrt{{0.1}^2+{0.5}^2}})=3.95\times {10}^{5}V$

Thus work done

$W=Q_o(V_B-V_A)$

$=5\times {10}^{-6}(3.95\times {10}^5-2.5\times {10}^5)$

$\Rightarrow W=0.725\approx 0.75J$