1. Electric Potential

Electric Potential

What Is Electric Potential?

Electric potential describes the electric potential energy available per unit charge at a particular location in an electric field.

It is represented by:

V

and defined as:

V = Eₚ/q

where:

• V = electric potential (V)
• Eₚ = electric potential energy (J)
• q = charge placed at that location (C)

Electric potential tells us how much electric potential energy each coulomb of charge would have at a particular position.

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The Volt

The SI unit of electric potential is the:

volt (V)

From:

V = Eₚ/q

we obtain:

1 V = 1 J/C

Therefore, a location at an electric potential of 10 V corresponds to:

10 J of electric potential energy per coulomb of charge

for a positive test charge.

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Electric Potential vs Electric Potential Energy

These two quantities are closely related but are not the same.

Electric potential

Depends on:

• source charges
• location in the electric field

It does not depend on the test charge used to examine the field.

Electric potential energy

Depends on:

• electric potential
• test charge

The relationship is:

Eₚ = qV

Therefore, two different charges placed at the same location experience the same electric potential but may have different electric potential energies.

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Example: Potential and Potential Energy

Suppose a location has:

V = 20 V

A charge:

q = 3 C

is placed there.

Use:

Eₚ = qV

Eₚ = 3 × 20

Eₚ = 60 J

If instead:

q = 0.5 C

then:

Eₚ = 0.5 × 20

Eₚ = 10 J

The electric potential remains 20 V, but the potential energy depends on the charge.

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Electric Fields and Electric Potential

An electric field describes the force that a positive test charge would experience per unit charge:

E = F/q

Electric potential describes potential energy per unit charge:

V = Eₚ/q

These quantities describe different aspects of the same electrical situation.

Electric field is related to how electric potential changes from one location to another.

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Direction of the Electric Field

The electric field points in the direction in which electric potential decreases most rapidly.

Therefore:

electric field direction: high potential → low potential

This is an extremely important relationship.

For a positive test charge, the electric force also acts in the direction of the electric field.

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Potential Difference

Usually, the most important quantity in a circuit or electric field is not the absolute potential at one point but the difference in potential between two points.

Potential difference is:

ΔV = V_B − V_A

It can also be related to the change in potential energy:

ΔEₚ = qΔV

Potential difference is commonly called:

voltage

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What Does a Potential Difference Mean?

Suppose two locations differ in potential by:

12 V

Since:

1 V = 1 J/C

a charge of 1 C experiences a potential-energy difference of:

12 J

between those locations.

For a charge of 2 C:

ΔEₚ = qΔV

ΔEₚ = 2 × 12

ΔEₚ = 24 J

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Work and Potential Difference

Moving charge through an electric potential difference involves energy transfer.

The magnitude of electrical work or energy transfer can be calculated using:

E = qV

or for a change between two points:

ΔEₚ = qΔV

This relationship connects:

• charge
• voltage
• energy

It is fundamental throughout electricity.

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Positive Charges and Potential

A positive charge naturally tends to move in the direction of the electric field if free to do so.

Since the field points:

high potential → low potential

a free positive charge tends to move toward:

lower electric potential

As it does so, its electric potential energy decreases.

The lost potential energy may become kinetic energy or another form of energy.

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Negative Charges and Potential

Negative charges behave differently.

Remember:

Eₚ = qV

For a negative charge, q is negative.

Therefore, a negative charge can decrease its potential energy by moving toward higher electric potential.

This means:

• positive charges tend toward lower potential
• negative charges tend toward higher potential

if only electrical forces act.

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Electric Potential Around a Positive Point Charge

Consider an isolated positive charge.

Its electric field points:

away from the charge

The electric potential is greatest near the positive charge and decreases as distance increases.

Far from the charge, the potential approaches zero if infinity is chosen as the zero reference.

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Electric Potential Around a Negative Point Charge

Around an isolated negative charge:

• electric field points toward the charge
• electric potential is negative
• potential becomes more negative closer to the charge
• potential approaches zero far away

Notice that electric potential can be:

• positive
• zero
• negative

depending on the source charges and chosen reference.

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Electric Potential of a Point Charge

For a point charge:

V = kQ/r

where:

• V = electric potential (V)
• k = Coulomb constant ≈ 8.99 × 10⁹ N·m²/C²
• Q = source charge (C)
• r = distance from the source charge (m)

Notice that Q keeps its sign.

Therefore:

positive Q → positive V

negative Q → negative V

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Potential Is a Scalar

Electric potential is a scalar quantity.

It has:

• magnitude
• sign

but no direction.

This is different from electric field, which is a vector quantity.

Therefore:

electric field → vector

electric potential → scalar

This makes calculations involving several charges much easier in many situations.

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Point Charge Example

Calculate the electric potential 0.50 m from a:

Q = +2.0 μC

point charge.

Convert:

Q = 2.0 × 10⁻⁶ C

Use:

V = kQ/r

V = (8.99 × 10⁹)(2.0 × 10⁻⁶)/0.50

V ≈ 35,960 V

Therefore:

V ≈ 3.6 × 10⁴ V

The positive sign occurs because the source charge is positive.

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Negative Charge Example

Calculate the potential 0.20 m from:

Q = −4.0 μC

V = kQ/r

V = (8.99 × 10⁹)(−4.0 × 10⁻⁶)/0.20

V ≈ −179,800 V

Therefore:

V ≈ −1.8 × 10⁵ V

The negative result indicates negative electric potential relative to the chosen zero at infinity.

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Effect of Distance

For a point charge:

V = kQ/r

Therefore:

V ∝ 1/r

If distance doubles:

r → 2r

then:

V → V/2

If distance triples:

r → 3r

then:

V → V/3

Electric potential decreases with distance according to an inverse-distance relationship.

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Potential vs Electric Field with Distance

For a point charge:

V ∝ 1/r

but:

E ∝ 1/r²

This means electric field strength decreases more rapidly with distance than electric potential.

For example, if distance doubles:

potential becomes:

1/2 as large

while field strength becomes:

1/4 as large

This distinction is important.

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Comparing Locations Around a Positive Charge

Suppose point A is:

r_A = 1 m

from a positive point charge.

Point B is:

r_B = 2 m

from the same charge.

Since:

V = kQ/r

we obtain:

V_A = kQ/1

and:

V_B = kQ/2

Therefore:

V_A = 2V_B

Point A has twice the electric potential of point B.

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Equipotential Lines

An equipotential line connects locations that have the same electric potential.

For a point charge, equipotential lines appear as concentric circles in a two-dimensional diagram.

In three dimensions, they form spherical surfaces.

Every point on the same equipotential has:

the same electric potential

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Equipotential Lines and Electric Fields

Electric field lines cross equipotential lines at:

90°

Therefore:

electric field lines are perpendicular to equipotential lines

Why?

Because moving along an equipotential requires no change in electric potential.

If the electric field had a component along the equipotential, it would produce a potential change.

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Moving Along an Equipotential

Suppose a charge moves from point A to point B along the same equipotential.

Then:

V_A = V_B

Therefore:

ΔV = 0

and:

ΔEₚ = qΔV

So:

ΔEₚ = 0

No net electrical work is required to move a charge between two points on the same equipotential surface.

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Closely Spaced Equipotential Lines

The spacing of equipotential lines tells us something about electric field strength.

Where equipotential lines are:

close together → stronger electric field

Where they are:

farther apart → weaker electric field

This is because a large potential change over a short distance corresponds to a strong electric field.

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Uniform Electric Fields

Between two large oppositely charged parallel plates, the electric field is approximately uniform away from the edges.

In a uniform field:

• field lines are parallel
• field strength is approximately constant
• equipotential lines are evenly spaced
• potential changes approximately linearly with distance

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Electric Field and Potential Difference

For a uniform electric field:

E = |ΔV|/d

where:

• E = electric field strength (V/m or N/C)
• |ΔV| = magnitude of potential difference (V)
• d = distance measured along the field direction (m)

Rearranging:

|ΔV| = Ed

This relationship is especially useful for parallel plates.

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Example: Parallel Plates

Two parallel plates have a potential difference of:

600 V

and are separated by:

0.030 m

The electric field strength is:

E = ΔV/d

E = 600/0.030

E = 20,000 V/m

or:

E = 2.0 × 10⁴ V/m

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Potential Between Parallel Plates

Suppose two plates are separated by:

0.10 m

with:

V_positive = 100 V

V_negative = 0 V

If the field is uniform, the potential decreases evenly with distance.

At halfway between the plates:

V = 50 V

At one-quarter of the distance from the 100 V plate:

V = 75 V

Potential changes linearly because the field is uniform.

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Electric Potential and Work

Suppose a positive charge moves from:

V_A = 80 V

to:

V_B = 20 V

Then:

ΔV = V_B − V_A

ΔV = 20 − 80

ΔV = −60 V

For:

q = +2.0 × 10⁻³ C

ΔEₚ = qΔV

ΔEₚ = (2.0 × 10⁻³)(−60)

ΔEₚ = −0.12 J

The negative result means the charge loses 0.12 J of electric potential energy.

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Potential Energy Becoming Kinetic Energy

If only the electric force acts:

decrease in electric potential energy → increase in kinetic energy

Therefore:

ΔK = −ΔEₚ

In the previous example:

ΔEₚ = −0.12 J

so:

ΔK = +0.12 J

The positive charge speeds up as it moves toward lower potential.

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Moving Against the Electric Field

To move a positive charge from lower potential to higher potential, an external force usually must perform work.

For example:

20 V → 80 V

represents:

ΔV = +60 V

For a positive charge:

ΔEₚ > 0

The charge gains electric potential energy.

This is similar conceptually to lifting an object against gravity.

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Gravitational Analogy

Electric potential can be compared with gravitational height.

For gravity:

higher position → greater gravitational potential energy for a positive mass

For electric fields:

electric potential plays a role somewhat similar to gravitational potential.

However, an important difference is that electric charge can be:

• positive
• negative

whereas ordinary mass is positive.

This causes negative charges to respond differently from positive charges.

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Potential from Several Charges

Because electric potential is a scalar, potentials from several point charges can simply be added:

V_total = V₁ + V₂ + V₃ + ...

For point charges:

V_total = k(Q₁/r₁ + Q₂/r₂ + Q₃/r₃ + ...)

Remember to include the sign of every charge.

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Example with Two Charges

At point P:

Q₁ = +2 μC at r₁ = 0.50 m

Q₂ = −1 μC at r₂ = 0.25 m

Potential from Q₁:

V₁ = kQ₁/r₁

V₁ = (8.99 × 10⁹)(2 × 10⁻⁶)/0.50

V₁ ≈ +35,960 V

Potential from Q₂:

V₂ = (8.99 × 10⁹)(−1 × 10⁻⁶)/0.25

V₂ ≈ −35,960 V

Therefore:

V_total = V₁ + V₂

V_total = 0 V

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Zero Potential Does Not Necessarily Mean Zero Electric Field

This is an important distinction.

In the previous example, the positive and negative potentials cancel because potential is scalar.

However, electric fields are vectors.

The electric fields at the same point might:

• reinforce
• partially cancel
• point in different directions

Therefore:

V = 0 does not automatically mean E = 0

Similarly:

E = 0 does not automatically mean V = 0

These are different physical quantities.

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Potential Between Equal Positive Charges

Consider two identical positive charges.

At the midpoint:

• their electric fields point in opposite directions
• their electric fields can cancel

Therefore:

E = 0

But each charge produces positive potential.

The potentials add:

V_total > 0

So this provides an example where:

E = 0 but V ≠ 0

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Potential Between Opposite Charges

Consider equal positive and negative charges.

At their midpoint:

the positive potential and negative potential can cancel.

Therefore:

V = 0

However, the electric fields between the charges point in the same general direction.

Therefore:

E ≠ 0

This provides an example where:

V = 0 but E ≠ 0

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Electric Potential Around Conductors

In electrostatic equilibrium, a conductor has the same electric potential throughout its interior and surface.

The conductor is therefore an:

equipotential

If different parts of the conductor had different potentials, free electrons would move until equilibrium was reached.

Inside a conductor in electrostatic equilibrium:

E = 0

and:

V = constant

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Electric Potential and Circuits

Voltage in an electrical circuit is a difference in electric potential.

A battery maintains a potential difference between its terminals.

For example:

a 9 V battery provides approximately:

9 J of energy per coulomb of charge

between its terminals under idealized conditions.

This connects electric-field concepts with circuit electricity.

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Batteries and Potential

A battery uses chemical processes to move charge in a way that maintains a potential difference.

The battery transfers energy to charge.

That charge can then transfer energy to components such as:

• lamps
• motors
• resistors
• electronic circuits

Therefore, voltage is fundamentally an energy-per-charge quantity.

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Electron Volts

For very small particles, joules can be inconveniently large units.

An electron volt (eV) is the energy gained by one elementary positive charge moving through a potential difference of 1 V.

1 eV = 1.602 × 10⁻¹⁹ J

For example, an electron accelerated through a potential difference of 100 V changes its kinetic energy in magnitude by:

100 eV

Electron volts are commonly used in:

• atomic physics
• nuclear physics
• particle physics

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Electric Potential Graphs

For a positive point charge:

V = kQ/r

As r increases:

V decreases toward zero.

The decrease is not linear.

Near the charge, potential changes rapidly with distance.

Farther away, it changes more gradually.

This corresponds to the fact that the electric field is stronger near the charge.

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Potential Gradient

Electric field strength is related to the potential gradient.

In one dimension:

E = −ΔV/Δx

for a sufficiently small interval or uniform field.

More generally:

E = −dV/dx

The negative sign means:

electric field points toward decreasing potential

A steep change in potential corresponds to a strong electric field.

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Comparing Two Locations

Suppose:

V_A = 500 V

V_B = 200 V

Then:

Point A has the higher electric potential.

The potential difference from A to B is:

ΔV = V_B − V_A

ΔV = 200 − 500

ΔV = −300 V

A positive charge naturally accelerated from A toward B would lose electric potential energy.

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Comparing Potential Does Not Require a Test Charge

Because electric potential describes the electrical environment itself, we can compare:

V_A and V_B

without specifying a test charge.

However, to determine the potential-energy difference, we need the charge:

ΔEₚ = qΔV

This distinction is important.

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Worked Example: Potential and Energy

A +5 μC point charge produces an electric potential at a location 0.20 m away.

Find the potential

V = kQ/r

V = (8.99 × 10⁹)(5 × 10⁻⁶)/0.20

V ≈ 2.25 × 10⁵ V

Now place a:

q = +2 nC

test charge there.

Convert:

q = 2 × 10⁻⁹ C

Potential energy:

Eₚ = qV

Eₚ = (2 × 10⁻⁹)(2.25 × 10⁵)

Eₚ ≈ 4.5 × 10⁻⁴ J

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Worked Example: Comparing Distances

A positive point charge produces:

V = 900 V

at:

r = 0.20 m

What is the potential at:

r = 0.60 m?

Since:

V ∝ 1/r

distance increases by a factor of:

0.60/0.20 = 3

Therefore potential becomes:

900/3

V = 300 V

No value for Q is required.

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Worked Example: Uniform Field

Two points are separated by:

d = 0.040 m

along a uniform electric field of:

E = 2500 V/m

Potential difference magnitude:

ΔV = Ed

ΔV = 2500 × 0.040

|ΔV| = 100 V

The point farther in the direction of the electric field is at the lower potential.

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Choosing the Correct Equation

Use:

V = Eₚ/q

when relating potential to potential energy and charge.

Use:

Eₚ = qV

when finding potential energy.

Use:

V = kQ/r

for the potential produced by a point charge.

Use:

V_total = V₁ + V₂ + ...

for several point charges.

Use:

E = |ΔV|/d

for a uniform electric field.

Use:

ΔEₚ = qΔV

when a charge moves between two potentials.

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Common Mistakes

Mistake: Confusing electric potential with electric potential energy.
Potential is energy per unit charge. Potential energy also depends on the test charge.

Mistake: Saying electric potential has a direction.
Electric potential is scalar.

Mistake: Treating electric potential as the same as electric field strength.
Potential is measured in volts; electric field strength is measured in N/C or V/m.

Mistake: Using V = kQ/r².
Potential follows 1/r, while electric field follows 1/r².

Mistake: Removing the sign of Q when calculating potential.
The sign of the source charge determines the sign of its potential.

Mistake: Saying electric field points from low potential to high potential.
The field points toward decreasing potential.

Mistake: Assuming zero potential means zero electric field.
V and E are different quantities.

Mistake: Assuming a positive and negative charge behave the same way in a potential difference.
Because Eₚ = qV, the sign of q matters.

Mistake: Saying work is required to move along an equipotential surface.
There is no change in electric potential or electric potential energy when moving along the same equipotential.

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Key Terms

Electric potential (V) — electric potential energy per unit charge at a location.

Volt (V) — unit of electric potential; 1 V = 1 J/C.

Electric potential energy (Eₚ) — energy associated with a charge's position in an electric field.

Potential difference — difference in electric potential between two locations.

Voltage — another name commonly used for electric potential difference.

Electric field — region in which electric charges experience electrical forces.

Point charge — idealized charge concentrated at a single location.

Equipotential line — line joining locations with equal electric potential.

Equipotential surface — surface on which electric potential is constant.

Potential gradient — rate at which electric potential changes with position.

Scalar — quantity having magnitude but no direction.

Electron volt (eV) — energy equal to 1.602 × 10⁻¹⁹ J.

Uniform electric field — field with approximately constant magnitude and direction.

Test charge — small charge used conceptually to investigate an electric field.

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Key Takeaways

• Electric potential is electric potential energy per unit charge.
• V = Eₚ/q.
• One volt equals one joule per coulomb.
• Electric potential and electric potential energy are different quantities.
• Potential energy is related to potential by Eₚ = qV.
• Potential difference describes the change in energy per unit charge between two locations.
• ΔEₚ = qΔV.
• Electric field points from higher potential toward lower potential.
• Positive charges naturally tend toward lower potential if free to move.
• Negative charges tend toward higher potential when only electrical forces act.
• For a point charge, V = kQ/r.
• Point-charge potential follows an inverse-distance relationship.
• Electric field follows an inverse-square relationship.
• Electric potential is scalar, while electric field is vector.
• Potentials from several charges can be added directly using their signs.
• Equipotential lines connect locations having equal potential.
• Electric field lines are perpendicular to equipotential lines.
• Closely spaced equipotential lines indicate a stronger electric field.
• In a uniform field, E = |ΔV|/d.
• Zero potential does not necessarily mean zero electric field.
• Voltage in electrical circuits is fundamentally an energy-per-charge concept.

Check Your Understanding

1. Define electric potential.
2. What does a potential of 1 V mean in terms of joules and coulombs?
3. Explain the difference between electric potential and electric potential energy.
4. A 4 C charge has 60 J of electric potential energy. Calculate the electric potential.
5. Calculate the potential energy of a +3 C charge at a location where V = 20 V.
6. In which direction does the electric field point relative to electric potential?
7. Describe how electric potential changes with distance from a positive point charge.
8. Describe how electric potential changes as you approach a negative point charge.
9. Write the equation for the potential produced by a point charge.
10. Calculate the potential 0.40 m from a +3.0 μC point charge.
11. If distance from a point charge doubles, what happens to electric potential?
12. If distance from a point charge doubles, what happens to electric field strength?
13. Explain why electric potential is easier to add than electric field when several charges are present.
14. What is an equipotential line?
15. Explain the relationship between equipotential lines and electric field lines.
16. A uniform electric field has strength 5000 V/m. Calculate the potential difference across a distance of 0.020 m along the field.
17. A +2 mC charge moves from 100 V to 40 V. Calculate its change in electric potential energy.
18. Explain why zero electric potential does not necessarily mean zero electric field.
19. Two identical positive charges are placed equal distances from a midpoint. Explain why the electric field can be zero there while the electric potential is not zero.
20. Explain how the concept of electric potential connects electric fields with the voltage used in electrical circuits.