Topic 0001: The Gradient of an Electrostatic Field

The voltage gradient of an electrostatic field is important, it is the the maximum rate of change of voltage with respect to distance in an electric field and has the units of volts/metre. The insulation strength of materials is defined in volts/metre and is important in the Low Voltage Directive (LVD) and the ESD (ElectroStatic Discharge) standard.

In this topic we develop a vector equation for the voltage gradient of an electrostatic field.

Figure 1 The Potential Difference Between Two Surfaces

Fig 1 shows two surfaces, S1 and S2 distance r1 and r2 from a positive charge q. The voltage (potential) on surface S1 is V1 and the voltage on surface S2 is V2.

The potential difference between the two surfaces is:-

$V_2 - V_1 = V_{21} {\text{ }}.....................................................................{\text{1}}$

Now assume that the distance between the two surfaces S2 and S1 is small. If the electric field strength is E, then

$\begin{gathered} \Delta V = E\Delta r \hfill \\ \frac{{\Delta V}} {{\Delta r}} = E \hfill \\ \frac{{\Delta V}} {{\Delta r}}{\text{ is called the gradient of the potential field}}................{\text{2}} \hfill \\ \end{gathered}$

In vector notation

$E = - {\mathbf{r}}\frac{{\partial V}} {{\partial r}}{\text{ }}.......................................................................{\text{3}}$

Where r is a unit vector in the direction of the radius r1

If the potential field is a function of x,y,z, that is V= F(x,y,z), we may write equation 3 in cartesian co-ordinates as:-

$\begin{gathered} E = - \left( {{\mathbf{i}}\frac{{\partial V}} {{\partial x}} + {\mathbf{j}}\frac{{\partial V}} {{\partial y}} + {\mathbf{k}}\frac{{\partial V}} {{\partial z}}} \right) \hfill \\ E = - \left( {{\mathbf{i}}\frac{\partial } {{\partial x}} + {\mathbf{j}}\frac{\partial } {{\partial y}} + {\mathbf{k}}\frac{\partial } {{\partial z}}} \right)V \hfill \\ E = - grad(V) \hfill \\ E = - \nabla V \hfill \\ \nabla = \left( {{\mathbf{i}}\frac{\partial } {{\partial x}} + {\mathbf{j}}\frac{\partial } {{\partial y}} + {\mathbf{k}}\frac{\partial } {{\partial z}}} \right){\text{ }}..................................................{\text{4}} \hfill \\ \nabla {\text{ is a vector operator pronounced del}} \hfill \\ \end{gathered}$

Where

i= a unit vector in the direction of the x axis

j= a unit vector in the direction of the y axis

k= a unit vector in the direction of the z axis

It is important to note that the electrical field E has the dimensions of volts/metre as well as newtons/coulomb. We shall frequently use volts/metre as the units of the electric field in this module. The insulation strength of material is specified in volts/metre.

The gradient we have developed above is along the radius vector and this is the maximum value it can have. The gradient of the electrical potential field is a vector quantity.

grad gives the maximum field strength at a point x,y,z. If we move along the surface S1 the gradient is everywhere zero and this is an equipotential surface. The surfaces S1 and S2 in Fig 1 should be thought of as spherical shells.

Note: Voltages can exist in any medium, such as free space, air, dielectric materials and conductors. The potential field and electric field are not confined to the conductors that make up the "circuit"

Author: Ron Hood

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 2.0 Licence.

Terms and conditions apply.