__Electric Potential Energy (U) and Electric Potential
(V): (____Notes from C. Erkal’s lectures PHYS 221____)__

Consider a parallel plate capacitor that produces a uniform electric field between its large plates. This is accomplished by connecting each plate to one of the terminals of a power supply (such as a battery).

Figure 1: An electric field is set up by the charged plates separated by a distance l. The charges on the plates are +Q and –Q.

Figure 2: An electric charge q is moved from point A towards point B with an external force T against the electric force qE.

Figure 3, 4: When it is moved through a distance d, its potential energy at the point B is qEd relative to the point A.

Figure 5: When released from B (T = 0), it will accelerate toward the lower plate. As it is moving toward the lower plate, its potential energy decreases and its Kinetic energy increases. When it reaches the lower plate (where we can choose the Potential energy to be zero), its potential energy at A is completely converted to Kinetic Energy at point B:

_{}

Note that qEd is the work done by the field as the charge moves under the force qE from B to A. Here m is the mass of the charge q, and v is its velocity as it reaches point A. Here we assumed that electric field is uniform! Work done by E field:

_{}

Let’s remember Kinetic Energy-Work theorem (Work Energy principle):

_{}

where we introduced the concept of potential energy and conservative force ( a force under which one can define a potential energy so that the work done only depends the differences of the potential energy function evaluated at the end points).

**A rule of thumb for deciding whether or not EPE is
increasing:**

If a charge is moving in the direction that it would
normally move, its electric potential energy is decreasing. If a charge is moved in a direction opposite
to that of it would normally move, its electric potential energy is increasing. This situation is similar to that of
constant gravitational field (g = 9,8 m/s^{2}). When you lift up an object, you are
increasing its gravitational potential energy.
Likewise, as you are lowering an object, its gravitational energy is
decreasing.

A General Formula for Potential Difference:

The work done by an E field as it act on a charge q to move it from point A to point B is defined as Electric Potential Difference between points A and B:

_{}

Clearly, the potential function V can be assigned to each
point in the space surrounding a charge distribution (such as parallel
plates). The above formula provides a
simple recipe to calculate work done in moving a charge between two points
where we know the value of the potential difference. The above statements and the formula are valid regardless of the
path through which the charge is moved.
A particular interest is the potential of a point-like charge Q. It can be found by simply performing the
integration through a simple path (such as a straight line) from a point A
whose distance from Q is r to infinity.
Path is chosen along a radial line so that _{} becomes simply Edr.
Since the electric field of Q is kQ/r^{2},

This process defines the electric potential of a point-like charge. Note that potential function is a scalar quantity as oppose to electric field being a vector quantity. Now, we can define the electric potential energy of a system of charges or charge distributions. Suppose we compute the work done against electric forces in moving a charge q from infinity to a point a distance r from the charge Q. The work is given by:

_{}

Note that if q is negative, its sigh should be used in the equation! Therefore, a system consisting of a negative and a positive point-like charge has a negative potential energy.

A negative potential energy means that work must be done against the electric field in moving the charges apart!

Now consider a more general case, which deals with the potential in the neighborhood of a number of charges as depicted in the picture below:

Let r_{1},r_{2},r_{3 }be the
distances of the charges to a field point A, and r_{12}, r_{13},
r_{23} represent the distance between the charges. The electric potential at point A is:

_{}

**Example:**

If we bring a charge Q from infinity and place it at point A the work done would be:

_{}

The total Electric Potential Energy of this system of charges namely, the work needed to bring them to their current positions can be calculated as follows: first bring q1 (zero work since there is no charge around yet), then in the field of q1 bring q2, then in the fields of q1 and q2 bring q3. Add all of the work needed to compute the total work. The result would be:

_{}

Finding Electric Field from Electric Potential:

The component of E in any direction is the negative of the rate of change of the potential with distance in that direction:

_{}

The symbol Ñ is called Gradient. Electric field is the gradient of electric potential. Electric field lines are always perpendicular to the equipotential surfaces.

**Equipotentail Surfaces:**

These are imaginary surfaces surrounding a charge
distribution. In particular, if the
charge distribution is spherical (point charge, or uniformly charged sphere),
the surfaces are spherical, concentric with the center of the charge
distribution. Electric field lines are
always perpendicular to the equipotential surfaces. The equation _{} implies that due to
the negative sign, the direction of E is opposite to the direction in which V
increases; E is directed from higher to lower levels of V (from higher
potential to lower potential). Another
words, the gradient of a scalar (in this case E field) is normal to a surface
of constant value (equipotential surface) of the scalar and in the direction of
maximum rate of change of constant scalar.
Remember this statement when we perform the experiment.