Q30P - Fundamentals Of Physics

Question

Figure 22-53 shows two concentric rings, of radii  R and,$\mathbf{3}\mathbf{.00}\mathit{R}$ that lie on the same plane. Point P lies on the central  Z 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 P is zero?

Step-by-Step Solution

Answer

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

Step 1: 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, $D\left(orz\right)=2R$.

 

Step 2: 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

Step 3: 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.$-4.19Q$