BE1-HEM1 Electromagnetics 1
Notes 2:Electric Potential
1. Electric potential difference VA − VB
From Notes 1, charge q placed in an electric field E experiences a force :-
Fel = qE
We stop the charge from accelerating by applying an equal and opposite force Fext = −Fel , and then
move the charge a short, straight displacement dl (dl is short enough that the electric force does not
change over its length). Depending on whether dl has a component against the electric force, or with
the electric force, we are doing work, or having work done to us respectively. The amount of work done
by us is calculated as the size of the force we apply×distance moved in the direction of that force.
Displacement moved against
electric force
q E
dl
The total work done by us is therefore:-
dWext = Fext .dl = −Fel .dl
We now move the charge a larger displacement on a path that is not necessarily a straight line, from an
arbitrary point B to an arbitrary point A, where the electric field may vary along the path. The path
comprises a sequence of small, straight displacements dl
B
dl
q E
A
the total work done by us is:-
Z A Z A
Wext = dWext = −q E.dl
B B
This is called a path integral, or line integral.
If we do work, the potential energy in the system increases - if work is done against us, the potential
energy decreases. The two determining factors are :-
- do we move with or against the electric force
- the sign of the charge q
The potential difference VA − VB is defined as the work done Wext when a charge is moved from B
to A, per unit charge:-
Z A Z B
Wext
VA − VB = =− E.dl = E.dl
q B A
1
, BE1-HEM1 Electromagnetics 1:Notes 2:Electric Potential
The potential difference VA − VB is the potential of A with respect to B and is often termed the voltage
of A with respect to B . If we move in the direction against E we move towards increasing values of
potential. If we move with E we move towards decreasing values of voltage.
V increases V decreases
E E
+q -q
As an example, we can find the potential difference between two points A and B on a radial path in the
electric field due to a point charge.
B
dr
E
A
+q
A rA Z rA
1 1
Z Z q q h q irA q
VA −VB = − E.dl = − r̂ . (r̂dr) = − dr = = −
B rB 4πǫr 2 rB 4πǫr
2 4πǫr rB 4πǫ rA rB
2. Conservative nature of electrostatic field
E B
dr θ
dl
A
+q
If we consider the path shown in the diagram, following the top of the path from A to B, the potential
difference between the two ends of dl is given by E.dl, which is |E|dl cos θ , or alternatively |E||dr|. The
potential difference is therefore only a function of the radial component of displacement - if we moved the
charge circumferentially, no work would be done as we would not be moving against or with the force.
This means that if we travelled around the entire loop from A to B and then back to A again, we would
do zero work in total as the radial position at the start and end of the loop would be the same rA . This
would apply to any closed loop chosen and is the definition of a conservative field:- WA−→A = 0. For
2
Notes 2:Electric Potential
1. Electric potential difference VA − VB
From Notes 1, charge q placed in an electric field E experiences a force :-
Fel = qE
We stop the charge from accelerating by applying an equal and opposite force Fext = −Fel , and then
move the charge a short, straight displacement dl (dl is short enough that the electric force does not
change over its length). Depending on whether dl has a component against the electric force, or with
the electric force, we are doing work, or having work done to us respectively. The amount of work done
by us is calculated as the size of the force we apply×distance moved in the direction of that force.
Displacement moved against
electric force
q E
dl
The total work done by us is therefore:-
dWext = Fext .dl = −Fel .dl
We now move the charge a larger displacement on a path that is not necessarily a straight line, from an
arbitrary point B to an arbitrary point A, where the electric field may vary along the path. The path
comprises a sequence of small, straight displacements dl
B
dl
q E
A
the total work done by us is:-
Z A Z A
Wext = dWext = −q E.dl
B B
This is called a path integral, or line integral.
If we do work, the potential energy in the system increases - if work is done against us, the potential
energy decreases. The two determining factors are :-
- do we move with or against the electric force
- the sign of the charge q
The potential difference VA − VB is defined as the work done Wext when a charge is moved from B
to A, per unit charge:-
Z A Z B
Wext
VA − VB = =− E.dl = E.dl
q B A
1
, BE1-HEM1 Electromagnetics 1:Notes 2:Electric Potential
The potential difference VA − VB is the potential of A with respect to B and is often termed the voltage
of A with respect to B . If we move in the direction against E we move towards increasing values of
potential. If we move with E we move towards decreasing values of voltage.
V increases V decreases
E E
+q -q
As an example, we can find the potential difference between two points A and B on a radial path in the
electric field due to a point charge.
B
dr
E
A
+q
A rA Z rA
1 1
Z Z q q h q irA q
VA −VB = − E.dl = − r̂ . (r̂dr) = − dr = = −
B rB 4πǫr 2 rB 4πǫr
2 4πǫr rB 4πǫ rA rB
2. Conservative nature of electrostatic field
E B
dr θ
dl
A
+q
If we consider the path shown in the diagram, following the top of the path from A to B, the potential
difference between the two ends of dl is given by E.dl, which is |E|dl cos θ , or alternatively |E||dr|. The
potential difference is therefore only a function of the radial component of displacement - if we moved the
charge circumferentially, no work would be done as we would not be moving against or with the force.
This means that if we travelled around the entire loop from A to B and then back to A again, we would
do zero work in total as the radial position at the start and end of the loop would be the same rA . This
would apply to any closed loop chosen and is the definition of a conservative field:- WA−→A = 0. For
2
