PHY C17: Electric Potential Energy

Let’s wrap up everything with know about electric fields:

• Electric Potential Energy
• Electric Potential
• Potential Difference
• Relationship between Electric Potential & Electric Field Strength
• Graphs

Let’s go!

What is electric potential energy?
The ability for an electric charge to do work in an electric field.

Specifically,
change in electric potential energy = work done by electric force in moving a charge between 2 points

Usually denoted as PE or Ep

Consider a charge in an electric field:

The charge will naturally move from point A to point B due to the electric force.
As it does so, it loses potential energy.
The difference in potential energy is the electric force x the distance AB = work done!

What is electric potential?
Work done per unit positive charge in bringing a test charge from infinity to a point.

It is denoted by V.

You might be familiar with that symbol V, since it also represents a similar value: potential difference or VOLTAGE. We’ll get to that later.

V is similar to gravitational potential, φ. We take infinity as having 0 potential since the electric force at an infinite distance from a field source is practically 0.

To move a charge to any position within an electric field, you must apply a force to that charge & travel a distance – you will do work & expend energy. The amount of work done per unit charge IS the electric potential.

How is this related to potential difference?
Potential difference is just a difference in electric potential between 2 points (duh).

When we say a battery has a voltage of 2V, we are really saying that the difference in electric potential between the terminals is 2V. Positive charges will travel from a high potential to a low one, so they will travel through a circuit from the + terminal to the – one!

How does electric potential relate to electric field strength?
For a specific point in an electric field:
Electric field strength = – potential gradient at that point

We’ve discussed this in the post about fields, but it’s worth repeating for electricity:
Since charges in an electric field have electric potential energy, they have the ability to do work by moving – more specifically, ACCELERATING.

Charges tend to change their position from a high potential to a low one. If you place a positive charge at a position with high electric potential, it starts to accelerate towards a position with lower electric potential. The cause of this mysterious acceleration is what we label as a FORCE. In fact, this is how forces arise – they stem from differences in potential between 2 points.

Specifically,
W = Fd
Change in potential energy = electric force x distance between 2 points
Electric force = change in potential energy/distance between 2 points

Dividing both sides by the charge of the object:
Electric field strength = potential difference/distance between 2 points
E = V/d

Wait a minute, we’ve seen that formula before: it’s the one we used in AS! There’s your proof.

In terms of calculus, we can look at the graphs of potential & field strength:

In a uniform field: Yellow: graph of field potential against distance, V = -kr + cPurple: graph of field strength against distance, E = k

The potential decreases linearly with distance, so the field strength is constant. Note that E = -dV/dr. Yellow: graph of field potential against distance, V = kQ/rPurple: graph of field strength against distance, E = kQ/r2

Both potential & field strength decrease with distance, at different rates. Note that E = -dV/dr.

To look for the field strength at a specific point, consider the potential difference between 2 points close together.

As we shrink the distance between them, the line showing their gradient becomes a TANGENT to that point. Thus, we can use differentiation to find the field strength!

However, we define E as being the force acting per unit positive charge. Thus, we must add a negative in front of it (as the potential gradient is negative but the force on positive charges is positive).

dV/dr = -E

How about graphs of field strength & potential INSIDE & OUTSIDE a spherical conductor?
Remember 2 facts about charged spherical conductors:

1. For points outside the sphere, the field from a spherical conductor can be treated as radiating from a POINT CHARGE at the centre of the sphere
2. Inside the sphere, there is 0 electric field

So, the graph of field strength would look like this:

Since field strength is 0, the electric potential MUST remain CONSTANT inside the sphere. In fact, the value of the electric potential inside the sphere is EQUAL to the electric potential at the surface of the sphere, because the field remains 0 from inside the sphere up to reach its surface. Thus, there must be no difference in potential from within the sphere & its surface.