Potential Energy

If you’ve studied gravitational potential energy, you already understand the core idea. Hold a ball above the ground — it has stored energy. Release it — the stored energy converts into motion as the ball falls.

Electric potential energy is the same concept, just with electric charges instead of gravity. When two charges are arranged in a particular configuration, there’s energy stored in that arrangement. Move the charges around, and you’re either storing more energy or releasing it. Same logic, different force — and conservation of energy still applies.

Building the Formula from What You Know

You learned in Section 5.2 that the force between two charges follows Coulomb’s law:

$\displaystyle F = \dfrac{kq_1 q_2}{r^2}$

Potential energy and force are closely related. In fact, if you know how force changes with distance, you can figure out potential energy (this involves calculus, which you don’t need for the MCAT). The result is:

Understanding the Sign: Positive vs. Negative Energy

The sign of U tells you something crucial about the system. This is where many students get confused, so let’s build intuition.

Like Charges: Positive Potential Energy

When both charges have the same sign (both positive, or both negative), the product $q_1 \times q_2$ is positive, so $U$ is positive.

What does positive energy mean physically? Think about it: like charges repel. To push them close together, you have to fight against that repulsion - you have to do work on the system. That work gets stored as potential energy, just like compressing a spring stores energy.

Unlike Charges: Negative Potential Energy

When the charges have opposite signs (one positive, one negative), the product $q_1 \times q_2$ is negative, so $U$ is negative.

What does negative energy mean? Unlike charges attract. They naturally “want” to come together - you don’t have to push them. In fact, you’d have to do work to pull them apart. The system is in an energy “hole” - it would take energy input to escape.

Quick Reference: What Does the Sign Mean?

How Does Energy Change as Charges Move?

The electric force is what physicists call a conservative force - the same category as gravity. This means we can track energy using potential energy, and total mechanical energy is conserved (as long as no other forces are doing work).

If you’ve studied conservation of energy, you know the key equation:

$\displaystyle K_1 + U_1 = K_2 + U_2$

(Initial kinetic energy + initial potential energy = final kinetic energy + final potential energy)

This means:

• When $U$ decreases, $K$ increases (potential energy converts to speed)
• When $U$ increases, $K$ decreases (speed converts to potential energy)

What Makes $U$ Increase or Decrease?

$U$ increases when you fight the natural tendency:

• Pushing like charges closer together (they want to repel)
• Pulling unlike charges apart (they want to attract)

$U$ decreases when you go with the natural tendency:

• Letting like charges fly apart (they naturally repel)
• Letting unlike charges come together (they naturally attract)

Force vs. Energy: Why One Has $r^2$ and the Other Has $r$

Here’s something that often confuses students: the force equation has $r^2$ in the denominator, but the energy equation only has $r$.

This means at large distances, you might barely feel any force between two charges (force has dropped to nearly zero), but there can still be significant potential energy stored in the system.

Classic Problem Type: Distance of Closest Approach

Here’s a common MCAT problem setup:

A positive charge is fired toward another positive charge (which is held fixed). How close does the moving charge get before it stops and bounces back?

Solution approach:

1. At the start: the moving charge has kinetic energy ($K_1 = \dfrac{1}{2}mv^2$) and some potential energy ($U_1$)
2. At closest approach: all the kinetic energy has converted to potential energy. The charge momentarily stops ($K_2 = 0$) before bouncing back
3. Apply conservation of energy: $K_1 + U_1 = U_2$

If the charges start very far apart, $U_1 \approx 0$, so:

$\displaystyle \dfrac{1}{2}mv^2 = \dfrac{kq_1 q_2}{r_{\text{closest}}}$

Solve for $r_{\text{closest}}$.

Energy in Uniform Electric Fields

When you have a uniform field (like between parallel plates), there’s a simpler formula for potential energy change:

$\displaystyle \Delta U = qEd$

where:

• $q$ = the charge being moved
• $E$ = the electric field strength
• $d$ = distance moved along (or against) the field direction

This is the electrical version of $\Delta U = mgh$ in gravity. Compare them: