Electric Potential and Energy

Overview

Electric Potential and Energy explains the energy viewpoint of Electric Fields. Instead of focusing only on force, we study how much work is needed to move charges and how energy changes in an electric field.

This topic is closely linked to Energy Forms and Conservation.

Definition

Electric potential at a point is the work done per unit positive charge by an external agent in bringing a small positive test charge from infinity to that point, without changing its kinetic energy.

$$V=\frac{W}{q}$$

where:

• $V$ = electric potential
• $W$ = work done by external agent
• $q$ = positive test charge

Units:

• volt (V)
• J C${}^{-1}$

Key Idea

Potential tells you the energy per unit charge at a point.

It is a scalar quantity.

Why It Matters

The potential viewpoint helps you:

• compare different locations in a field without dealing with force components each time
• connect electric fields to energy change and particle speed
• interpret equipotential diagrams
• avoid mixing vector and scalar quantities

Key Representations

Why Infinity Is Used as Reference

For isolated charges, electric potential decreases with distance and approaches zero far away.

Hence we define:

$$V=0\text{at infinity}$$

This gives a convenient reference level.

Potential Due to a Point Charge

For source charge $Q$:

$$V=\frac{1}{4\pi {\epsilon }_0}\frac{Q}{r}$$

where:

• $r$ = distance from the charge

Sign of Potential

• positive charge gives positive potential
• negative charge gives negative potential

Trend

$$V\propto \frac{1}{r}$$

So potential decreases more gradually than field strength.

Superposition of Potential

If several charges are present:

$$V_{\text{net}}=V_1+V_2+V_3+\cdots$$

Because potential is scalar, add algebraically with signs.

This differs from electric field, which must be added vectorially.

Electric Potential Energy

Potential energy of charge $q$ at potential $V$:

$$U=qV$$

For point-charge interaction:

$$U=\frac{1}{4\pi {\epsilon }_0}\frac{Qq}{r}$$

Units:

• joule (J)

Meaning of the Sign of Potential Energy

Like Charges

$Qq>0$

$$U>0$$

Energy must be supplied to bring them closer.

Unlike Charges

$Qq<0$

$$U<0$$

The system releases energy when brought together.

Change in Potential Energy

If a charge moves between two points:

$$\Delta U=q\Delta V$$

where:

$$\Delta V=V_f-V_i$$

Work Done: Field vs External Agent

This is a common source of confusion.

Work Done by External Agent

For slow controlled movement with no kinetic-energy change:

$$W_{\text{ext}}=\Delta U$$

Work Done by Electric Field

$$W_{\text{field}}=-\Delta U$$

If the field does positive work, potential energy decreases.

Energy Conservation View

If only electric forces act:

$$\Delta K+\Delta U=0$$

So:

• losing potential energy increases kinetic energy
• gaining potential energy decreases kinetic energy

This is useful in particle-acceleration problems.

Equipotential Lines and Surfaces

An equipotential joins points with the same electric potential.

Properties

• no work is done moving a charge along it
• it is always perpendicular to electric field lines
• closer spacing means stronger field

Examples

Point Charge

Concentric circles in 2D or spheres in 3D.

Uniform Field

Parallel lines perpendicular to the field direction.

Relation Between Potential and Field

Electric field strength is the potential gradient:

$$E=-\frac{dV}{dx}$$

Meaning:

• the electric field points toward lower potential
• a steeper drop in potential means a stronger field

For a uniform field between plates:

$$E=\frac{V}{d}$$

where $d$ is plate separation.

Potential vs Potential Energy

Electric Potential $V$

• property of the location in the field
• independent of the test charge
• unit: V or J C${}^{-1}$

Potential Energy $U$

• depends on both the location and the charge placed there
• depends on the sign and magnitude of $q$

$$U=qV$$

Typical Interpretations

Positive Charge Released Freely

Moves from higher potential to lower potential.

Negative Charge Released Freely

Moves opposite to the field direction, often toward higher potential.

But Remember

Potential energy depends on charge sign, not just potential alone.

Graph Trends with Distance from a Point Charge

Potential

$$V\propto \frac{1}{r}$$

Potential Energy

$$U\propto \frac{1}{r}$$

Sign depends on $Qq$.

Field Strength Comparison

$$E\propto \frac{1}{r^2}$$

Common Mistakes

1. Confusing potential with potential energy
2. Forgetting the sign of charge in $U=qV$
3. Using vector rules for potential
4. Assuming higher potential always means higher potential energy
5. Forgetting the infinity reference
6. Mixing work done by the field with work done by the external agent

Quick Exam Method

If Asked for Potential at a Point

Use:

$$V=\sum \frac{1}{4\pi {\epsilon }_0}\frac{Q}{r}$$

If Asked for Energy Change

Use:

$$\Delta U=q\Delta V$$

If Asked for Speed Gain

Use:

$$\Delta K=-\Delta U$$

Summary

Core equations:

$$V=\frac{W}{q}$$ $$V=\frac{1}{4\pi {\epsilon }_0}\frac{Q}{r}$$ $$U=qV$$ $$U=\frac{1}{4\pi {\epsilon }_0}\frac{Qq}{r}$$ $$\Delta U=q\Delta V$$ $$E=-\frac{dV}{dx}$$

Links

• Electric Fields
• Charged Particles in Fields
• Electric Fields Common Exam Traps
• Gravitational vs Electric Fields
• Gravitational Fields
• Energy Forms and Conservation