Figure 22-53 shows two concentric rings, of radiiRand,$\mathbf{3}\mathbf{.00}\mathit{R}$ that lie on the same plane. Point Plies on the centralZ axis, at distance $\mathit{D}\mathbf{}\mathbf{=}\mathbf{}\mathbf{2}\mathbf{.00}\mathit{R}$from the center of the rings. The smaller ring has uniformly distributed charge$\mathbf{+}\mathit{Q}$. In terms of Q, what is the uniformly distributed charge on the larger ring if the net electric field at Pis zero?

1. Two concentric rings of radii, R and 3R lie on the same plane, where the smaller ring has a charge of$+Q$uniformly distributed.
2. A point P lies on the central z-axis at a distance,role="math" $D\left(orz\right)=2R$.

Using the concept of the electric field of a charged rod, we can get the net electric field of the distribution at the center of the formed circle.

Formula:

The electric field at a point due to a ring,$E=\frac{1}{4\pi {\epsilon }_o}\frac{qz}{{(z^2+R^2)}^{\frac{3}{2}}}$ (i)

where, R = radius of the ring

z = distance of the field point on the axis of the ring

Let, the charge on the larger ring is. Thus, the net electric field at point P is given using equation (i), which is equated to zero as follows:

$\begin{array}{l}\frac{1}{4\pi {\epsilon }_o}\frac{qz}{{(z^2+R^2)}^{\frac{3}{2}}}+\frac{1}{4\pi {\epsilon }_o}\frac{qz}{{(z^2+{\left(3R\right)}^2)}^{\frac{3}{2}}}=0\\ q=-Q{\left(\frac{13}{5}\right)}^{\frac{3}{2}}\\ =-4.19Q\end{array}$

Hence, the value of the charge of the larger ring is.$-4.19Q$