4 Engineering Example 1

4.1 Electrostatic charge

Introduction

Electrostatic charge is important both where it is wanted, as in the electrostatic precipitator plate systems used for cleaning gases, and where it is unwanted, such as when charge builds up on moving belts. This Example is concerned with a charged object with a particular idealised shape - a sphere. However, similar analytical calculations can be carried out for certain other shapes and numerical methods can be used for more complicated shapes.

The electric field at all points inside and outside a charged sphere is given by

$E\left(r\right)=\frac{Qr}{4\pi {\epsilon }_0{a}^3}\text{if}r<a$ (1a)

$E\left(r\right)=\frac{Q}{4\pi {\epsilon }_0{r}^2}\text{if}r\ge a$ (1b)

where ${\epsilon }_0$ is the permittivity of free space, $Q$ is the total charge, $a$ is the radius of the sphere, and $r$ is the radial distance between the centre of the sphere and a point of observation (see Figure 2).

Figure 2 :

The electric field associated with electrostatic charge has a scalar potential. The electric field defined by (1a) and (1b) shows only a radial dependence of position. Therefore, the electric scalar potential $V\left(r\right)$ is related to the field $E\left(r\right)$ by

$E\left(r\right)=-\frac{dV}{dr}.$ (2)

Problem in words

A sphere is charged with a uniform density of charge and no other charge is present outside the sphere in space. Determine the variation of electric potential with distance from the centre of the sphere.

Mathematical statement of problem

Determine the electric scalar potential as a function of $r,V\left(r\right)$ , by integrating (2).

Mathematical analysis

Equation (2) yields $V\left(r\right)$ as the negative of the indefinite integral of $E\left(r\right)$ .

$-\int \; <!--nolimits\; -->dV=\int \; <!--nolimits\; -->E\left(r\right)dr.$ (3)

Using (1a) and (1b) with (3) leads to

$V\left(r\right)=-\frac{Q}{4\pi {\epsilon }_0{a}^3}\int \; <!--nolimits\; -->rdr\text{if}r<a$ (4a)

$V\left(r\right)=-\frac{Q}{4\pi {\epsilon }_0}\int \; <!--nolimits\; -->\frac{dr}{r^2}\text{if}r\ge a$ (4b)

Using the facts that $\int \; <!--nolimits\; -->rdr=r^2∕2+c_1\text{and}\int \; <!--nolimits\; -->\frac{dr}{r^2}=-\frac{1}{r}+c_2,$

(4a) and (4b) become

$V\left(r\right)=-\frac{Q{r}^2}{8\pi {\epsilon }_0{a}^3}+c_1\text{if}r<a$ (5a)

$V\left(r\right)=\frac{Q}{4\pi {\epsilon }_{0}r}+c_2\text{if}r\ge a$ (5b)

The integration constant $c_2$ can be determined by assuming that the electric potential is zero at an infinite distance from the sphere:

$\underset{r\to \infty }{lim}\left[V\left(r\right)\right]=0$ $\Rightarrow \underset{r\to \infty }{lim}\left[-\frac{Q}{4\pi {\epsilon }_{0}r}\right]+c_2=0$ $\Rightarrow c_2=0.$

The constant $c_1$ can be determined by assuming that the potential is continuous at $r=a.$

From equation (5a)

$V\left(a\right)=-\frac{Q{a}^2}{8\pi {\epsilon }_0{a}^3}+c_1$

From equation (5b)

$V\left(a\right)=\frac{Q}{4\pi {\epsilon }_{0}a}$

Hence

$c_1=\frac{Q}{4\pi {\epsilon }_{0}a}+\frac{2Q}{8\pi {\epsilon }_{0}a}=\frac{3Q}{8\pi {\epsilon }_{0}a}.$

Substituting for $c_1$ in (5), the electric potential is obtained for all space is:

$V\left(r\right)=\frac{Q}{4\pi {\epsilon }_0}\left(\frac{3a^2-r^2}{2a^3}\right)\text{if}r<a.$

$V\left(r\right)=\frac{Q}{4\pi {\epsilon }_{0}r}\text{if}r\ge a$

Interpretation

The potential of the electrostatic field outside a charged sphere varies inversely with distance from the centre of the sphere. Inside the sphere, the electrostatic potential varies with the square of the distance from the centre.

An Engineering Exercise in HELM booklet  29.3 derives the corresponding expressions for the variation of the electrostatic field and an Engineering Exercise in HELM booklet  27.4 calculates the potential energy due to the charged sphere.

Exercises

1. Find $\int \; <!--nolimits\; -->\left(2x-{\text{e}}^x\right)dx$
2. Find $\int \; <!--nolimits\; -->3{\text{e}}^{2x}dx$
3. Find $\int \; <!--nolimits\; -->\frac{1}{3}\left(x+cos2x\right)dx$
4. Find $\int \; <!--nolimits\; -->7x^{-2}dx$
5. Find $\int \; <!--nolimits\; -->{\left(x+3\right)}^{2}dx$ , (be careful!)

Answer