2.4 Electric potential energy

Electric potential energy describes the energy stored in an electric field due to the arrangement of charges. It connects the geometry of charge configurations to the forces and energy transfers that govern how charged particles move, how circuits operate, and how devices like capacitors store energy.

Definition of electric potential energy

Electric potential energy is the energy a charged particle has because of its position in an electric field. Think of it like gravitational potential energy: a ball held above the ground has stored energy due to its position in a gravitational field. Similarly, a charge placed near other charges has stored energy due to its position in an electric field.

This concept is central to analyzing electric circuits, capacitors, and interactions between charged particles at every scale.

Relation to electric field

The electric field and electric potential energy are tightly linked. The electric field points in the direction of steepest decrease in potential energy. Mathematically, the electric field is the negative gradient of the potential energy (per unit charge).

This means if you know how potential energy varies across a region, you can determine the force a charge will experience at any point in that region.

Units of measurement

• Electric potential energy is measured in joules (J)
• For atomic-scale problems, the electron volt (eV) is more convenient: 1 eV = $1.6 \times 10^{-19}$ J. This is the energy gained by a single electron moving through a potential difference of 1 volt.
• Voltage (V) measures potential energy per unit charge, so 1 V = 1 J/C

Work done by electric forces

When a charged particle moves through an electric field, the field exerts a force on it and does work. The work-energy theorem tells you that this work equals the change in the particle's kinetic energy, which corresponds to an equal and opposite change in potential energy.

Conservative nature of electric force

The electric force is a conservative force. This means:

• The work it does on a charge depends only on where the charge starts and ends, not on the route it takes
• Total mechanical energy (kinetic + potential) is conserved in a system of charges with no external forces
• You can define a potential energy function that depends only on position

This is the same property that makes gravity conservative. Friction, by contrast, is not conservative because the work it does depends on the path.

Path independence

Because the electric force is conservative, calculating the work done between two points is straightforward. You don't need to trace the actual trajectory of the charge. Pick any convenient path between the starting and ending positions, and the work (and therefore the change in potential energy) will be the same.

This simplifies problems enormously, especially with complicated charge arrangements.

Potential energy in uniform fields

A uniform electric field has the same strength and direction everywhere. You encounter this most often between the plates of a parallel plate capacitor, where the field lines are straight and evenly spaced.

Linear variation with position

In a uniform field, potential energy changes linearly with displacement along the field direction:

$\Delta U = qEd$

where $q$ is the charge, $E$ is the field strength, and $d$ is the displacement parallel to the field. On a graph of $U$ vs. position, this relationship produces a straight line whose slope is proportional to the electric field magnitude.

Analogy to gravitational potential energy

This setup mirrors gravitational potential energy near Earth's surface. Compare the two formulas:

• Gravitational: $\Delta U = mgh$
• Electrical: $\Delta U = qEd$

Mass $m$ corresponds to charge $q$, gravitational acceleration $g$ corresponds to electric field strength $E$, and height $h$ corresponds to displacement $d$. If you're comfortable with gravitational PE, the uniform-field electrical case works the same way.

Potential energy of point charges

Point charges are the building blocks for more complex charge distributions. Understanding the potential energy between two point charges lets you analyze everything from atomic structure to molecular bonding.

Coulomb's law and potential energy

The potential energy between two point charges is:

$U = k\frac{q_1 q_2}{r}$

where $k$ is Coulomb's constant ($8.99 \times 10^9 \, \text{N·m}^2/\text{C}^2$) and $r$ is the distance between the charges.

Pay attention to the sign:

• Like charges (both positive or both negative): $U > 0$. You had to do work to push them together, so energy is stored.
• Opposite charges: $U < 0$. The charges attract, and you'd have to do work to pull them apart.

Also note that $U$ is inversely proportional to $r$, so the interaction is strongest at short distances and weakens as charges move apart.

Multiple charge systems

For a system with more than two charges, use the superposition principle: add up the potential energy of every unique pair.

For example, with three charges ($q_1, q_2, q_3$), the total potential energy is:

$U_{\text{total}} = k\frac{q_1 q_2}{r_{12}} + k\frac{q_1 q_3}{r_{13}} + k\frac{q_2 q_3}{r_{23}}$

Be careful not to double-count pairs. For $N$ charges, there are $\frac{N(N-1)}{2}$ unique pairs.

Potential energy vs kinetic energy

The interplay between potential and kinetic energy determines how charged particles speed up, slow down, and change direction in electric fields.

Conservation of energy principle

In an isolated electrostatic system (no friction, no radiation losses), total energy is conserved:

$K_i + U_i = K_f + U_f$

This is one of the most powerful tools in electrostatics. If you know a particle's initial speed and position, you can find its speed at any other position without needing to track forces and accelerations along the way.

Conversion between forms

• A positive charge released in an electric field accelerates in the field direction, converting potential energy into kinetic energy.
• A positive charge moving against the field slows down, converting kinetic energy back into potential energy.
• The rate of conversion depends on the field strength and the particle's charge.

Equipotential surfaces

Equipotential surfaces are imaginary surfaces where every point has the same electric potential (and therefore the same potential energy per unit charge). They're a visualization tool that reveals the structure of an electric field.

Definition and properties

• A charge moving along an equipotential surface experiences no change in potential energy, so the electric field does zero work on it.
• Equipotential surfaces are always perpendicular to electric field lines at every point.
• For a point charge, the equipotential surfaces are concentric spheres. For a uniform field, they're parallel planes.

Relation to electric field lines

• Electric field lines cross equipotential surfaces at right angles.
• Where equipotential surfaces are packed closely together, the electric field is strong. Where they're spread apart, the field is weak.
• No work is done moving a charge along an equipotential surface, which is why conductors in electrostatic equilibrium have surfaces that are equipotential.

Applications of electric potential energy

Capacitors and energy storage

Capacitors store energy in the electric field between their plates. The energy stored is:

$U = \frac{1}{2}CV^2$

where $C$ is the capacitance and $V$ is the voltage across the plates. This energy can be released quickly, which is why capacitors are used in camera flashes, defibrillators, and power supply smoothing circuits.

Particle accelerators

Particle accelerators exploit electric potential energy to speed up charged particles. A particle with charge $q$ moving through a potential difference $\Delta V$ gains kinetic energy equal to $q \Delta V$. By chaining together many stages of potential difference, accelerators push particles to enormous speeds for use in physics research, cancer radiation therapy, and materials analysis.

Potential energy in electric circuits

Electric potential energy is what drives current through a circuit. Charges flow from regions of high potential energy to low potential energy, and components in the circuit convert that energy into other forms (light, heat, motion).

Batteries as potential energy sources

A battery uses chemical reactions to maintain a potential difference (voltage) between its terminals. This potential difference pushes charges through the circuit. Real batteries have internal resistance, which means some of the stored energy is lost as heat inside the battery itself, reducing the voltage available to the external circuit.

Potential difference across components

The voltage drop across a circuit component represents the potential energy lost by charges as they pass through it. Kirchhoff's voltage law states that the sum of all voltage gains and drops around any closed loop in a circuit equals zero. This is a direct consequence of energy conservation. Power dissipated by a resistor is calculated as $P = IV$, where $I$ is current and $V$ is the voltage across the resistor.

Calculation methods

Integration of electric field

For non-uniform fields, you calculate the change in potential energy by integrating the electric field along a path:

$\Delta U = -q\int_{a}^{b} \mathbf{E} \cdot d\mathbf{l}$

Here's how to use this in practice:

1. Identify the start point $a$ and end point $b$
2. Choose a convenient path between them (any path works, since the force is conservative)
3. Write out the electric field $\mathbf{E}$ as a function of position along your chosen path
4. Evaluate the dot product $\mathbf{E} \cdot d\mathbf{l}$ and integrate
5. Multiply by $-q$ to get the change in potential energy

Superposition principle for multiple charges

To find the total potential energy of a system of discrete charges:

1. List every unique pair of charges
2. Calculate $U = k\frac{q_i q_j}{r_{ij}}$ for each pair
3. Sum all the pairwise energies

For symmetric charge distributions, look for pairs that are identical in magnitude and distance to speed up the calculation.

Potential energy curves

Potential energy curves plot $U$ as a function of position or separation distance. They give you a visual way to understand forces, equilibrium, and stability without solving equations.

Interpretation and analysis

• The slope of the curve at any point gives the force (with a negative sign): $F = -\frac{dU}{dx}$. A steep slope means a strong force; a flat region means near-zero force.
• Where the slope is zero, the net force is zero, and the particle is in equilibrium.
• The curve's shape tells you whether a particle will oscillate, escape, or remain trapped.

Stable vs unstable equilibrium points

• Stable equilibrium occurs at a local minimum of the potential energy curve. If you nudge the particle slightly, it experiences a restoring force that pushes it back. Think of a ball sitting at the bottom of a bowl.
• Unstable equilibrium occurs at a local maximum. A small nudge causes the particle to accelerate away from that point. Think of a ball balanced on top of a hill.

Identifying these points on a potential energy curve is a common exam question. Look for where the slope is zero, then check the curvature: concave up = stable, concave down = unstable.