Question

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?

Short answer

The uniformly distributed charge on the larger ring if the net field at P has zero value is$-4.19Q$ .

Step-by-step solution

The given data

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" $D\left(orz\right)=2R$.

Understanding the concept of electric field 

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

Calculation of the charge of the larger 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.$-4.19Q$