2 Engineering Example 1

2.1 Electrostatic potential

Introduction

The electrostatic potential due to a point charge [maths rendering] coulombs at a position [maths rendering] (m) from the charge is given by

[maths rendering]

where [maths rendering] , the permittivity of free space, [maths rendering] and [maths rendering] .

The field strength at position [maths rendering] is given by [maths rendering] .

Problem in words

Find the electric field strength at a distance of 5 m from a source with a charge of 1 coulomb.

Mathematical statement of the problem

[maths rendering]

Substitute for [maths rendering] and [maths rendering] and use [maths rendering] , so that

[maths rendering]

We need to differentiate [maths rendering] in order to find the electric field strength from the relationship [maths rendering]

Mathematical analysis

[maths rendering]

When [maths rendering] ,

[maths rendering] ( [maths rendering] .)

Interpretation

The electric field strength is [maths rendering] at [maths rendering] m.

Note that the field potential varies with the reciprocal of distance (i.e. inverse linear law with distance) whereas the field strength obeys an inverse square law with distance.