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Electric Field and Potential — Physics STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 SciencePhysicsElectric Field and Potential5 Marks
Question
Consider a circular ring of radius r, uniformly charged with linear charge density $\lambda.$ Find the electric potential at a point on the axis at a distance x from the centre of the ring. Using this expression for the potential, find the electric field at this point.
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Answer


Radius = r
So, $2\pi\text{r}=$ Circumference
Charge density $=\lambda$
Total charge $=2\pi\text{r}\times\lambda$
Electric potential $=\frac{\text{Kq}}{\text{r}}$
$=\frac{1}{4\pi\in_0}\times\frac{2\pi\text{r}\lambda}{\big(\text{x}^2+\text{r}2\big)^{\frac{1}{2}}}$
$=\frac{\text{r}\lambda}{2\in_0\big(\text{x}^2+\text{r}^2\big)^{\frac{1}{2}}}$

So, Electric field $=\frac{\text{V}}{\text{r}}\cos\theta$
$=\frac{\text{r}\lambda}{2\in_0\big(\text{x}^2+\text{r}^2\big)^{\frac{1}{2}}}\times\frac{\text{x}}{\big(\text{x}^2+\text{r}^2\big)^{\frac{1}{2}}}$
$=\frac{\text{r}\lambda\text{x}}{2\in_0\big(\text{x}^2+\text{r}^2\big)^{\frac{3}{2}}}$

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