# Episode 408: Field strength and potential energy

This episode introduces the above three quantities for the electric field. The students’ familiarity with the equivalent concepts in the gravitational field should help here. The major difference is that because of repulsion, by defining the zero of potential to be at infinity, we can have positive potentials and positive potential energies in the electric field, whereas they are always negative in the (solely attractive) gravitational field.

**Summary**

- Discussion: Field strength (5 minutes)
- Worked examples: Field strength (10 minutes)
- Discussion: Potential energy and potential (15 minutes)
- Demonstration: Potential around a charged sphere (30 minutes)
- Discussion: Field strength and potential gradient (10 minutes)
- Worked examples: The non-uniform electric field (25 minutes)

**Discussion: Field strength**

Recall: how is field strength defined at a point in the gravitational field? (As the force per unit mass placed at that point in the field – with units therefore of N kg^{-1}.)

What would therefore be the natural way to extend this definition to the electric field? (As the force per unit charge. Thus it would have units of N C^{-1}.)

We thus define the electric field strength at a point in a field as:

*E* = *F/Q* where *E* = electric field strength (N C^{-1})

*F* = force on charge *Q* at that point if the field

**Important notes:**

- The field strength is a property of the
**field**and not the particular charge that is placed there. For example, at a point where the field strength is 2000 N C^{-1}, a 1 C charge would feel a force of 2000 N whereas a 1 mC charge would feel a force of 2 N; the same field strength, but different forces due to different charges. - The field strength is a vector quantity. By convention, it points in the direction that a positive charge placed at that point in the field would feel a force.

As will be explained in the next episode, the unit for electric field strength can also be expressed as volts per metre, V m^{-1}.

Now, for the non-uniform field due to a point (or spherical) charge, we can use Coulomb’s law to find an expression for the field strength. Consider the force felt by a charge q in the field of another charge Q, where the charges are separated by a distance r:

by Coulomb’s Law.

But *E* = *F*/*q* and so

This is our result for the field strength at a distance *r* from a (point or spherical) charge *Q*.

**Worked examples: Field strength**

Episode 408-1: Field strength (Word, 31 KB)

**Discussion: Potential energy and potential**

We now turn to considerations of energy. Again, just as in the gravitational case, we choose to define the zero of electric potential energy at infinity. However, because of the existence of repulsion, we have the possibility of positive potential energy values as well as negative ones.

Consider bringing a positive charge *q* from infinity towards a fixed, positive charge *Q*. Because of the repulsion between the charges, we must do work on *q* to bring it closer to *Q*. This work is stored as the electric potential energy of *q*. The same would apply if both charges were negative, due to their mutual repulsion. In both cases therefore, the potential energy of *q* increases (from zero) as it approaches *Q*; i.e. electric potential energy of charge q is positive.

If *Q* and *q* are of opposite signs, however, they attract each other, and now it would take work to separate them. This work is stored as the electric potential energy of *q*, and so *q’s* potential energy increases (toward zero) as their separation increases; i.e. *q’s* potential energy is negative.

With the aid of integration, we can use Coulomb’s law to find the electrical potential energy of *q* in the field of *Q*. The final expression turns out as:

*EPE* = k*Qq*/*r*

where *r* is the separation of the charges. *EPE* is measured in joules, J.

Notes on this expression:

By considering the work done if *q* were fixed and *Q* were brought up from infinity, it should be clear that this expression is the *EPE* of either charge in the other’s field.

This expression is valid for the non-uniform field around point or spherical charges.

By defining potential energy to be zero at infinity, this expression gives us absolute values of potential energies. Differences can we worked out as change in

*EPE* = k*Qq* (1/*r*_{2} – 1/*r*_{1}).

It is essential to include the sign of the charges in this expression to get the correct *EPE*. Thus two positive or two negative charges yield positive *EPE* values, whereas opposite charges yield negative *EPE* values, as we would expect from our discussion above.

We can now go on to discuss potential. In the gravitational field, potential at a point was defined as the potential energy per unit mass at that point. The natural extension to electric fields is therefore as the potential energy per unit charge:

*V* = *EPE* / *Q*

*V* is therefore measured in joules per coulomb, J C^{-1}, which has the alternative, and more familiar name of volts, V. (Students sometimes worry about the fact the potential has the symbol *V* and its unit is also V for volts – the context makes it clear which V is being used, but be aware of this possibility for confusion).

Note again that potential is a property of the field and not the individual charge placed there. Thus at a point in a field where the potential is 500 V, a 1 mC charge has a potential energy of 0.5 J whereas a 1 C charge has a potential energy of 500 J. (One might say that potential is to potential energy exactly as field strength is to force in the field).

For the non-uniform field around a point charge, the above expression for *EPE* gives us a simple expression for potential. The potential at a distance *r* from a point or spherical charge *Q* is given by:

Note that this gives positive or negative values for potential depending upon whether *Q* is positive or negative. (Remember though that a negative charge placed where there is a negative potential will have a positive potential energy as expected due to the repulsion of the two negative charges).

For the non-uniform field around a point charge, the above expression for EPE gives us a simple expression for potential. The potential at a distance r from a point or spherical charge Q is given by:

Note that this gives positive or negative values for potential depending upon whether Q is positive or negative. (Remember though that a negative charge placed where there is a negative potential will have a positive potential energy as expected due to the repulsion of the two negative charges).

**Demonstration: Potential around a charged sphere**

Exploring the electrical potential near a charged sphere. This makes use of a flame probe; it is essential to practise the use of this before demonstrating in front of a class.

Episode 408-2: Potential near a charged sphere (Word, 102 KB)

Episode 409-2: Flame probe construction (Word, 237 KB)

**Discussion: Field strength and potential gradient**

As with the gravitational field there is a deep and important connection between the rate at which potential changes in a field, and the field strength there. Quite simply, the field strength is equal to the negative of the rate of change of potential:

*E* = – (change in *EPE*)/ (change in distance), or

The minus sign is again a relic of the more precise vector equation – it is there to give the direction of the field strength. The diagram below shows a plot of potential around a positive charge, and the two gradients drawn show where the field is high and where it is low. (Note that both of these gradients are negative. This would give a positive field strength by *E* = -d*V*/d*r*, indicating that the field acts in the positive *x*-direction, as indeed it does. This is what was meant by the minus sign being a relic of the more precise vector equation).

We can now also understand why it is that a conductor is an equipotential surface. Inside a conductor, no electric field can exist – if it did, the charges would feel a force and move around in such a way as to reduce the field. At some point equilibrium is reached and the field is zero. If the field is zero, then the potential gradient must be zero – i.e. the conductor is an equipotential surface.

It should now also be clear why equipotential surfaces get further apart as the field decreases in strength as was seen in the diagrams in episode 1, and why field strength can be quoted in units of V m^{-1}.

**Worked examples: The non-uniform electric field**

Episode 408-3: Non-uniform electric fields (Word, 34 KB)

**Download this episode**

Episode 408: Field strength and potential energy (Word, 179 KB)