Question

A Van de Graaff generator is a device for generating a large electric potential by building up charge on a hollow metal sphere. A typical classroom-demonstration model has a diameter of $30cm$.

a. How much charge is needed on the sphere for its potential to be$500,000V$?

b. What is the electric field strength just outside the surface of the sphere when it is charged to $500,000V$?

Short answer

a. The temperature is$Q=\mathbf{830}\mathbf{}\mathit{n}\mathit{C}$

b. There is a $\mathbf{3}\mathbf{.}\mathbf{3}\mathbf{\times }{\mathbf{10}}^{\mathbf{6}}\mathbf{}\mathit{N}\mathbf{/}\mathit{C}$electric field.

Step-by-step solution

Given information and theory used

Given :

Diameter of the classroom-demonstration model : $30cm$.

a. Potential to be on the sphere : $500,000V$

b. The sphere is charged to : $500,000V$

Theory used :

The potential of the spherical conductor is given by$V=\frac{1}{4\pi {\epsilon }_0}\frac{Q}{R}$

Calculating the charge needed and electric field strength 

a. Here $R=15cm$is the radius of the sphere, is the potential of the spherical conductor. This gives

$Q=4\pi {\epsilon }_{0}RV=\boxed{\mathbf{830}\mathbf{}\mathit{n}\mathit{C}}$for the charge.

b. The field is $E=\frac{1}{4\pi {\epsilon }_0}\frac{Q}{R^2}=\underset{V}{\underbrace{\frac{1}{4\pi {\epsilon }_0}\frac{Q}{R}}}\frac{1}{R}=\frac{V}{R}$

As a result,

$E=\boxed{\mathbf{3}\mathbf{.}\mathbf{3}\mathbf{\times }{\mathbf{10}}^{\mathbf{6}}\mathbf{}\mathit{N}\mathbf{/}\mathit{C}}$.