Electric Potential

A region in space exists around a charged object, Q, called an electric field, E (in N/C).

A positive test charge, q, that finds itself in this field will have a force, F, acting on it.

Further away, the positive test charge has less force acting on it.

Energy is the ability to do work (force acting over a distance) and there is clearly electrical energy (potential energy) available to do work on the positive test charge q. The potential energy at A is greater than at B because the force is greater. The force depends upon the magnitude of the charge of q, so it is useful to describe the potential energy inside an electric field as the potential energy per unit charge. This is called electric potential (or simply potential when “electric” is understood).

The unit for electric potential is the volt (V), which is equal to a Joule / Coulomb (J/C).

The electric potential at a point, r, in an electric field around a charge Q is given by the equation:

From this equation it can be seen that the electric potential is greater at A than at B (because you divide by r the distance from Q). And as the test charge q is moved further away from Q, the electric potential becomes less and less. By definition, the electric potential will be zero at infinity.

Potential Difference - a.k.a.Voltage

Consider the change in potential energy as the charge q moves from A to B:

• A positive charge loses potential energy when it moves in the direction of the electric field.
• A negative charge gains potential energy when it moves in the direction of the electric field.

Only differences in potential energy are physically measurable, so it is often more useful to discuss the potential difference or voltage between two points in an electric field.

Consider two parallel plates with opposite charges. The test charge q will experience a force moving it from the positive to the negative plate. For a positive test charge the positive plate is at a high potential and the negative plate is at a low potential. The situation is reversed for a negative test charge.

Potential difference, the voltage between A and B, is the change in energy per Coulomb by the charge q when moved from A to B or the negative of the work per Coulomb needed to move the charge from B to A.

definition of Voltage

• Voltage is negative for a positive charge moving from high(+) to low(-) potential.
• Voltage is positive for a negative charge moving from high(-) to low(+) potential.

Electric Potential Between Two Parallel Plates

In a uniform electric field, like the one that exists between parallel plates, there exists a relationship between electric potential and electric field.

Consider a test charge in between two parallel plates, as shown in the diagram to the right.  If we don’t worry about signs, then the work done on the charge as it moves between the plates is equal to the charge multiplied by the voltage between the plates:

Work is also equal to the force times the distance moved, W = Fd. And the force is equal to the electric field strength times the charge, F = qE. Therefore:

Lines of Equipotential

The electric potential can be represented diagrammatically by drawing equipotential lines or, in three dimensions, equipotential surfaces. An equipotential surface is one on which all points are at the same potential—that is the potential difference between any two points on the surface is zero. An equipotential surface must be perpendicular to the electric field at every point. If this were not the case then as you move a charge from one point to another you would move against or with an electric field and change the potential of the charge. On an equipotential surface, no work is required to move a charge from one point to the other.

The ElectronVolt

Another unit of energy other than the Joule that is useful for dealing with the energies associated with electrons, protons, and atoms, is the electron volt, eV, defined as the energy acquired by a particle carrying the charge of an electron (q = e) as a result of being accelerated through a potential difference of 1V.

Since ΔPE = qV,

definition of the electronVolt

Examples

9. Suppose an electron in the picture tube of a television set is accelerated from rest through a potential difference (voltage) of 5000V.
   1. What is the change in potential energy of the electron?
   2. What is the speed of the electron as a result of this acceleration?
   3. Repeat for a positive charge moving from A to B.
      
10. Two horizontal parallel plates are 0.0300m apart and the electric potential difference between them is 100.0V. A deuterium nucleus is initially at rest near the positive plate. Determine
    1. the work done by the electric force on the particle as it passes between the plates
    2. the particle’s kinetic energy just before it strikes the negative plate.
11. Determine the potential at a point P to a point charge of magnitude 3.00μC if point P is 0.400m from the charge. Determine the work which must be done in order to move a 0.100μC charge from infinity to point P.
12. Two point charges (Q₁ = 3.00μC, Q₂ = -2.50μC) are placed 0.050m apart, as shown in the diagram. Determine the potential at the point p halfway between the two charges.
    

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