Advanced Higher Physics

Electric Potential

Electric Potential 1

V = work done / q (measured in J C-1)

Defined as ‘the work done per unit positive charge in moving a charge from infinity to a point in an electrical field’

Gives the definition of the volt -

one volt (1 V) = one joule per coulomb (1 JC-1)

Electric Potential 2

The potential difference (p.d.) between two points A and B is the work done per unit charge in moving between points A and B.

d

B A

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( also,

* This is only true for a UNIFORM field. )

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E =V

d

Electric Potential 3

Law of conservation of energy tells us that work done in moving charge from point A to point B is independent of the route taken.

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If the p.d. between A and B is V, the same amount of work must be done in moving a unit of charge from A to B, whatever path is taken.

This is because the electric field is a conservative field.

Electric Potential 4

From the equation E = V / d we can see that electrical field strength, E, can be expressed volts per metre, V m-1

E can be thought of as a ‘potential gradient’

In a non-uniform field,

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E = −dV

dx

Electric Potential 5

using r to represent distance,

But, we already know that -

Combining these gives -

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Electric Potential 6

Integrating as shown, we get

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Electric Potential 7

NB - be careful with the sign of the potential.

In moving a positive charge from infinity to r, the charge will have gained potential energy, as work has to be done on the charge against the electric field.

Electric Potential 8

So if we have defined the potential to be zero at infinity, the potential V must be positive for all r less than infinity. Thus the potential at r is given by -

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Electric Potential 9

Unlike the electric field, the electric potential around a point charge decays as 1 / r , not 1 / r2.

The potential is a scalar quantity, not a vector quantity, although its sign is determined by the sign of the charge Q.

Electric Potential 10

The field strength and potential around a positive point charge are plotted below

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Electric Potential 11

We can plot the equipotential lines (connecting points of the same potential) around a point charge, as shown below

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The equipotentials form a set of concentric spheres around the charge

Note that the equipotentials cut the electric field lines at right angles. There is no work done in moving a charged particle along an equipotential.