Q38P - Fundamentals Of Physics

Question

Figure 22-58a shows a circular disk that is uniformly charged. The central z axis is perpendicular to the disk face, with the origin at the disk. Figure 22-58b gives the magnitude of the electric field along that axis in terms of the maximum magnitude $E_m$ at the disk surface. The z axis scale is set by${\mathit{z}}_{\mathbf{s}}\mathbf{=}\mathbf{ }\mathbf{8}\mathbf{.0}\mathbf{ }\mathbf{\text{cm}}$. What is the radius of the disk?

Step-by-Step Solution

Answer

The radius of the disk is.$6.93\text{\hspace{0.17em}cm}$

Step 1: The given data

• A circular disk of uniform charge.
• The central z-axis is perpendicular to the face of the disk.
• Maximum magnitude of the electric field is.$E_m$
• The z-axis scale set is.$z_s=8\text{\hspace{0.17em}cm}$

Step 2: Understanding the concept of electric field

Using the concept of the electric field at a point from the disk along its central axis, we can get the ratio of the electric field at a point and maximum electric field, thus solving it by substituting the given data, we can get the radius of the disk.

 

Formula:

The magnitude of the electric field produced by the disk at a point on its central axis,                          $E=\frac{\sigma }{2{\epsilon }_o}\left(1-\frac{z}{\sqrt{z^2+R^2}}\right)$                                                  (i)

where,  = surface charge density

z = distance on the central axis of the disk  

R = Radius of the disk

Step 3: Calculation of the disk

Using equation (i) for the electric field at a point from the disk and maximum electric field (at$z=0$), we can get that 

$\begin{array}{c}\frac{E}{E_{max}}=1-\frac{z}{\sqrt{z^2+R^2}}\\ \frac{1}{2}=1-\frac{z}{\sqrt{z^2+R^2}}\text{ (}\because \text{from graph,}\frac{E}{E_{max}}=\frac{1}{2}\text{)}\\ \frac{z}{\sqrt{z^2+R^2}}=\frac{1}{2}\\ 3z^2=R^2\\ R=z\sqrt{3}\\ =4\times \sqrt{3}\text{\hspace{0.17em}cm }(\because z=4,fromgraph)\\ =6.93\text{\hspace{0.17em}cm}\end{array}$

Hence, the value of the radius of the disk is .$6.93\text{\hspace{0.17em}cm}$