# 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:

The potential decreases linearly with distance, so the field strength is constant. Note that E = -dV/dr.

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: