Electric Potential and Electric Potential Energy

Whenever a charged object sits near another charge, it is affected by the surrounding electric field. That field can do work on the charge, and this leads to two closely related but different quantities: electric potential energy and electric potential.

Electric potential energy, \(U\), is the stored energy a charge has because of its position in an electric field, similar to how gravitational potential energy depends on height. Electric potential, \(V\), is the electric potential energy per unit charge at a point in space. Potential belongs to the field itself, while potential energy belongs to a specific charge placed in that field.

For two point charges \(q_1\) and \(q_2\) separated by a distance \(r\), the electric potential energy of the system is:

\(U = \dfrac{k q_1 q_2}{r}\)

Here \(k \approx 8.99 \times 10^9\ \)N\( \cdot \)m\(^2/\)C\(^2\) is Coulomb's constant, the same constant used in the electric force formula. In fact, \(U\) is the work an external agent must do to bring the charges from an infinite distance apart to a separation \(r\). If the charges have the same sign, \(U\) is positive (they repel, so energy is stored by pushing them together). If the charges have opposite signs, \(U\) is negative (they attract each other).

Worked example: potential energy of two charges

Find the electric potential energy of two charges, \(q_1 = 3.0 \times 10^{-6}\ \)C\(\) and \(q_2 = -2.0 \times 10^{-6}\ \)C\(\), separated by \(r = 0.50\ \)m\(\).

\(U = \dfrac{k q_1 q_2}{r} = \dfrac{(8.99 \times 10^9)(3.0 \times 10^{-6})(-2.0 \times 10^{-6})}{0.50}\)

\(U \approx -0.108\ \)J\(\)

The negative sign shows the charges attract, so energy would be released if they were allowed to move closer together.

The electric potential created by a single point charge \(q\) at a distance \(r\) is the potential energy per unit of test charge placed there:

\(V = \dfrac{U}{q_{test}} = \dfrac{k q}{r}\)

Electric potential is measured in volts (V), where \(1\ \)V\( = 1\ \)J/C\(\). Unlike the electric field, which is a vector, electric potential is a scalar quantity, it has a size but no direction, which often makes it easier to add contributions from multiple charges: you simply add the potentials algebraically, keeping track of sign.

Worked example: potential from a point charge

What is the electric potential at a distance of \(0.20\ \)m\(\) from a charge of \(q = 5.0 \times 10^{-9}\ \)C\(\)?

\(V = \dfrac{k q}{r} = \dfrac{(8.99 \times 10^9)(5.0 \times 10^{-9})}{0.20} \approx 225\ \)V\(\)

In most real problems you care about the change in potential between two points, called the potential difference or voltage:

\(\Delta V = V_B - V_A = \dfrac{\Delta U}{q}\)

Rearranged, this gives the work done when a charge \(q\) moves through a potential difference \(\Delta V\):

\(W = q \, \Delta V = -\Delta U\)

This is why a 9-volt battery, for example, does 9 joules of work on every coulomb of charge that flows through it. For a uniform electric field, such as between two parallel plates a distance \(d\) apart, potential difference simplifies further:

\(\Delta V = E d \quad \)or equivalently\( \quad E = \dfrac{\Delta V}{d}\)

This links potential directly back to the electric field: the field points from high potential toward low potential, and its strength equals how quickly the potential changes with distance.

Worked example: potential difference and work

A proton, \(q = 1.6 \times 10^{-19}\ \)C\(\), moves through a potential difference of \(\Delta V = 120\ \)V\(\). Find the change in potential energy and the work done on the proton.

\(\Delta U = q \, \Delta V = (1.6 \times 10^{-19})(120) = 1.92 \times 10^{-17}\ \)J\(\)

\(W = -\Delta U = -1.92 \times 10^{-17}\ \)J\(\)

The field does positive work on the proton (it accelerates in the direction of decreasing potential), so the proton's potential energy decreases by \(1.92 \times 10^{-17}\ \)J\(\).

Electric potential energy tells you how much work it takes to assemble a set of charges, while electric potential tells you what that field would do to any charge placed at a given point, independent of what that charge is. Once you know the potential at two points, the difference between them, \(\Delta V\), tells you exactly how much energy a moving charge will gain or lose, which is the foundation for understanding circuits, capacitors, and the motion of charged particles in electric fields.

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