# Electric Potential and Potential Difference

"The outcome of any serious research can only be to make two questions grow where one question grew before"
Thorstein Veblen

• The field near a system of charges can also be described by a scalar quantity known as the "Electric Potential".

" A potential difference of one volt exists between two points when one Joule of work is required to move one Coulomb of charge from one point to the other"

Between two points A and B we may write

WAB  =  -VAB q

where VAB = VB - VA  is the potential difference between A and B.

Note that WAB is the work done by the electric field in moving the charge.  The work done by the "external agent"  is -WAB.
• Units of potential difference are volts

1 Volt  =  1 Joule/Coulomb (J/C)

• In a region of space where there is an electric field the work done by the electric field, dW, when a positive point charge, q, is displaced by ds is given by,
Therefore,

For a uniform electric field we obtain,

where an arbitrary path can always be split into sections along E and sections perpendicular to E.

Note that this means that the electric field can be expressed in the units V/m.  [ 1 N/C = 1V/m ]
The electric field is a conservative field
• This means that the potential difference between two points is independent of the path taken.  Every point in space has a single value of V and E.

Food for thought....

The gravitational field behaves in exactly the same way.  Changes in gravitational energy are independent of the path taken.  Climbing stairs from one floor to another involve the same amount of work against gravity as riding an elevator.
Note that in simple gravitational applications we don't usually define a gravitational potential only gravitational potential energy (mgh).  In this case the gravitational potential is defined as gh.

• Exactly equivalent to gravity, it is CHANGES in potential difference, , which are defined.  To obtain absolute values of V physicists usually define V = 0 at infinity.  But this is an arbitary definition; in engineering applications it is often convenient to define the earth as V = 0.

• Potential due to a point charge

For a single point charge Q the potential difference between A and B is given by,

where E is the field due to a point charge and ds = dr , so that,

If we assume rB= ∞ then VB = 0 and,

Note that the potential is inversely proportional to r, rather than r2 as in the case of the electric field.
• Since V is a scalar the potential due to multiple point charges is found by adding the potential due to the individual charges (taking into account the sign of the charge).
• Continuous Charge Distributions
For continuous distributions of charge we may write,

The electric potential due to a continuous charge distribution can be calculated in a similar manner to the electric field due to such a distribution.  For example the potential at point P due to a uniform ring of charge (below).

• Equipotential Surfaces

An Equipotential Surface is a surface in space on which all points have the same potential.  Since all points have the same potential it requires no work to move a charge on such a surface.  This means that there is no component of E in the plane of the surface, in other words E must be at right angles to the surface.
Electric Field Lines and Equipotential Surfaces are at right angles
(Red dotted lines below)

In the period that Einstein was active as a professor, one of his students came to him and said: "The questions of this year's exam are the same as last years!" "True," Einstein said, "but this year all answers are different."
Albert Einstein

Dr. C. L. Davis
Physics Department
University of Louisville
email: c.l.davis@louisville.edu