Electric Potential Energy

What Is Electric Potential Energy?

Electric potential energy is energy stored because of the positions of electric charges relative to one another in an electric field.

It is usually represented by:

Eₚ or U

and is measured in:

joules (J)

Electric potential energy depends on:

• the amount of charge
• the signs of the charges
• the positions of the charges
• the electric potential at the location

A charge can gain or lose electric potential energy as it moves through an electric field.

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Electric Potential Energy and Electric Potential

Electric potential energy and electric potential are closely related, but they are not the same quantity.

Electric potential energy

Electric potential energy describes the energy associated with a particular charge at a particular location.

Unit:

joule (J)

Electric potential

Electric potential describes electric potential energy per unit charge at a location.

Unit:

volt (V)

Their relationship is:

Eₚ = qV

where:

• Eₚ = electric potential energy (J)
• q = charge (C)
• V = electric potential (V)

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Electric Potential Is a Property of the Location

Suppose a particular point in an electric field has:

V = 100 V

That electric potential exists whether we place a test charge there or not.

If we place:

q = +1 C

there:

Eₚ = qV

Eₚ = 1 × 100

Eₚ = 100 J

If instead:

q = +3 C

then:

Eₚ = 3 × 100

Eₚ = 300 J

The location still has the same potential, but the charge has a different potential energy.

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The Sign of the Charge Matters

Because:

Eₚ = qV

the sign of q affects the electric potential energy.

At:

V = +100 V

a charge:

q = +2 C

has:

Eₚ = (+2)(+100)

Eₚ = +200 J

But a charge:

q = −2 C

has:

Eₚ = (−2)(+100)

Eₚ = −200 J

Therefore, positive and negative charges behave differently in the same electric potential.

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Potential Difference and Changes in Potential Energy

When a charge moves between two locations:

ΔV = V_f − V_i

The corresponding change in electric potential energy is:

ΔEₚ = qΔV

or:

ΔEₚ = q(V_f − V_i)

where:

• ΔEₚ = change in electric potential energy
• q = moving charge
• V_f = final electric potential
• V_i = initial electric potential

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Interpreting the Sign of ΔEₚ

If:

ΔEₚ > 0

the charge has gained electric potential energy.

If:

ΔEₚ < 0

the charge has lost electric potential energy.

If:

ΔEₚ = 0

there has been no change in electric potential energy.

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Example: Positive Charge

A:

+2.0 C

charge moves from:

V_i = 50 V

to:

V_f = 150 V

First:

ΔV = 150 − 50

ΔV = +100 V

Then:

ΔEₚ = qΔV

ΔEₚ = (+2.0)(+100)

ΔEₚ = +200 J

The positive charge gains:

200 J

of electric potential energy.

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Example: Positive Charge Moving to Lower Potential

A:

+3.0 C

charge moves from:

200 V

to:

50 V

Potential difference:

ΔV = 50 − 200

ΔV = −150 V

Then:

ΔEₚ = (+3.0)(−150)

ΔEₚ = −450 J

The charge loses:

450 J

of electric potential energy.

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Positive Charges and Electric Fields

Electric field direction is defined as the direction of force on a positive test charge.

Electric fields point:

higher potential → lower potential

Therefore, when a positive charge moves naturally in the direction of the electric field:

electric potential decreases

and:

electric potential energy decreases

The lost electric potential energy can be converted into kinetic energy.

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Negative Charges Behave Differently

For a negative charge:

q < 0

Therefore, its potential energy behaves oppositely with electric potential.

A negative charge moving toward higher electric potential can actually have decreasing electric potential energy.

For example:

q = −2 C

moves from:

20 V → 80 V

Then:

ΔV = +60 V

ΔEₚ = (−2)(+60)

ΔEₚ = −120 J

The negative charge loses 120 J of electric potential energy even though it moved to a higher electric potential.

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Direction of Natural Motion

If released from rest and acted on only by the electric force:

Positive charge

tends to move:

high V → low V

Negative charge

tends to move:

low V → high V

In both cases, spontaneous motion caused by the electric force tends to move the charge toward:

lower electric potential energy

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Electric Potential Energy and Work

When the electric field does work on a charge:

W_field = −ΔEₚ

Therefore:

if electric potential energy decreases, the electric field does positive work.

If:

ΔEₚ = −50 J

then:

W_field = +50 J

The electric field has transferred 50 J of energy from electric potential energy into another form.

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External Work

If an external force moves a charge slowly against the electric force, the external force can increase the system's electric potential energy.

For an ideal slow movement:

W_external = ΔEₚ

For example, if moving a charge requires:

20 J

of external work against the electric force, the electric potential energy increases by:

20 J

assuming no other energy transfers.

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Gravitational Analogy

Electric potential energy can be compared with gravitational potential energy.

If you lift an object:

external work → increased gravitational potential energy

If the object falls:

gravitational potential energy → kinetic energy

Similarly, separating or rearranging electric charges can store electric potential energy.

When charges are released:

electric potential energy → kinetic energy

However, there is an important difference:

electric charges can be positive or negative

while ordinary gravitational mass is positive.

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Like Charges

Like charges repel.

Consider two positive charges.

To push them closer together, an external force must work against their repulsion.

Therefore:

separation decreases → electric potential energy increases

If released, the charges accelerate apart.

Their electric potential energy decreases while their kinetic energy increases.

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Opposite Charges

Opposite charges attract.

Consider a positive charge and a negative charge.

To pull them apart, an external force must work against their attraction.

Therefore:

separation increases → electric potential energy increases

If released, opposite charges naturally accelerate toward each other.

Their electric potential energy decreases.

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Potential Energy of Two Point Charges

For two point charges:

Eₚ = kQq/r

where:

• Eₚ = electric potential energy (J)
• k = 8.99 × 10⁹ N·m²/C²
• Q and q = charges (C)
• r = separation between charges (m)

The signs of both charges must be included.

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Two Positive Charges

If:

Q > 0

and:

q > 0

then:

Qq > 0

Therefore:

Eₚ > 0

Two like positive charges have positive electric potential energy relative to infinite separation.

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Two Negative Charges

If:

Q < 0

and:

q < 0

then:

Qq > 0

Therefore:

Eₚ > 0

Two negative charges also have positive potential energy relative to infinite separation.

The important feature is that the charges have the same sign.

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Opposite Charges

If:

Q > 0

and:

q < 0

then:

Qq < 0

Therefore:

Eₚ < 0

Opposite charges have negative electric potential energy relative to infinite separation.

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Why Can Potential Energy Be Negative?

The zero of electric potential energy is often defined when charges are infinitely far apart.

For opposite charges, energy must be supplied to separate them completely.

Their bound arrangement therefore has less energy than the zero-energy reference at infinity.

That is why:

Eₚ < 0

is physically meaningful.

Negative potential energy does not mean "negative energy exists" in a problematic sense. It means the system has less energy than the chosen reference state.

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Example: Two Positive Charges

Two charges are:

Q = +2.0 μC

q = +3.0 μC

separated by:

r = 0.50 m

Use:

Eₚ = kQq/r

Eₚ = (8.99 × 10⁹)(2.0 × 10⁻⁶)(3.0 × 10⁻⁶)/0.50

Eₚ ≈ 0.108 J

Therefore:

Eₚ ≈ +0.108 J

The positive sign is expected because the charges have the same sign.

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Example: Opposite Charges

Suppose:

Q = +2.0 μC

q = −3.0 μC

r = 0.50 m

Then:

Eₚ = (8.99 × 10⁹)(2.0 × 10⁻⁶)(−3.0 × 10⁻⁶)/0.50

Eₚ ≈ −0.108 J

The negative sign is expected because the charges have opposite signs.

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How Potential Energy Changes with Distance

For two point charges:

Eₚ = kQq/r

Therefore, the magnitude of potential energy follows:

|Eₚ| ∝ 1/r

As separation increases, the potential energy approaches:

0

when the charges become extremely far apart.

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Like Charges Moving Apart

For like charges:

Eₚ is positive.

As they move farther apart:

Eₚ decreases toward zero

If they are free to move, their lost electric potential energy can become kinetic energy.

Therefore:

electric PE decreases → kinetic energy increases

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Opposite Charges Moving Together

For opposite charges:

Eₚ is negative.

As they move closer:

Eₚ becomes:

more negative

This also represents a decrease in electric potential energy.

The decrease can become kinetic energy.

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Electric Potential Energy in a Uniform Field

Between parallel charged plates, the electric field can be approximately uniform.

For a charge moving through a potential difference:

ΔEₚ = qΔV

If the uniform field is known:

|ΔV| = Ed

so the magnitude of the potential-energy change can be related to:

|ΔEₚ| = |q|Ed

The signs and direction of motion must still be considered carefully.

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Example Between Parallel Plates

A positive charge:

q = +4.0 × 10⁻⁶ C

moves through a potential decrease of:

250 V

Therefore:

ΔV = −250 V

Use:

ΔEₚ = qΔV

ΔEₚ = (4.0 × 10⁻⁶)(−250)

ΔEₚ = −1.0 × 10⁻³ J

The charge loses:

1.0 mJ

of electric potential energy.

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Equipotential Lines and Potential Energy

An equipotential line connects points having the same electric potential.

If a charge moves along an equipotential:

ΔV = 0

Therefore:

ΔEₚ = qΔV

and:

ΔEₚ = 0

The charge's electric potential energy does not change.

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Conservation of Energy

One of the most important applications of electric potential energy is the conservation of energy.

If only the electric force acts:

Eₚ,i + K_i = Eₚ,f + K_f

where:

• Eₚ = electric potential energy
• K = kinetic energy

Therefore:

ΔK = −ΔEₚ

A decrease in electric potential energy produces an equal increase in kinetic energy.

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Charge Released from Rest

Suppose a charge begins at rest.

Then:

K_i = 0

If it loses:

50 J

of electric potential energy:

ΔEₚ = −50 J

Then:

ΔK = +50 J

Therefore:

K_f = 50 J

The charge's electric potential energy has been converted into kinetic energy.

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Example: Finding Kinetic Energy

A:

+5.0 × 10⁻⁶ C

charge is released from rest and moves through a potential drop of:

2000 V

Potential-energy change:

ΔEₚ = qΔV

ΔEₚ = (5.0 × 10⁻⁶)(−2000)

ΔEₚ = −0.010 J

Therefore:

ΔK = +0.010 J

Since it started from rest:

K_f = 0.010 J

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Finding Speed from Potential Difference

Kinetic energy is:

K = ½mv²

If a particle starts from rest and loses electric potential energy:

½mv² = −ΔEₚ

For an appropriate potential difference:

½mv² = |qΔV|

Therefore:

v = √(2|qΔV|/m)

This equation can be used to find the speed of charged particles accelerated through a known potential difference when non-relativistic physics is appropriate.

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Example: Accelerating a Charged Particle

A particle has:

q = +2.0 × 10⁻⁶ C

m = 0.004 kg

and is accelerated from rest through a potential drop of:

100 V

Energy gained:

K = |qΔV|

K = (2.0 × 10⁻⁶)(100)

K = 2.0 × 10⁻⁴ J

Now:

K = ½mv²

2.0 × 10⁻⁴ = ½(0.004)v²

v² = 0.10

v ≈ 0.316 m/s

Therefore:

v ≈ 0.32 m/s

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Electrons Accelerated Through a Voltage

Electrons are commonly accelerated using electric fields.

Although electrons are negative, the energy gained by an electron accelerated through a potential difference of magnitude V can be written:

K gained = e|ΔV|

where:

e = 1.602 × 10⁻¹⁹ C

If an electron is accelerated through:

100 V

then it gains:

100 eV

of kinetic energy.

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The Electron Volt

An electron volt (eV) is a unit of energy.

It is defined as the energy change of one elementary charge moving through a potential difference of 1 V.

1 eV = 1.602 × 10⁻¹⁹ J

Therefore:

1 keV = 1000 eV

1 MeV = 10⁶ eV

1 GeV = 10⁹ eV

Electron volts are especially useful in atomic, nuclear, and particle physics.

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Conservation of Energy with External Work

If an external force also does work, mechanical energy of the charge-field system does not necessarily remain constant by itself.

A useful energy relationship is:

W_external = ΔK + ΔEₚ

when other energy transfers are absent.

If a charge is moved slowly so its kinetic energy does not change:

ΔK = 0

Therefore:

W_external = ΔEₚ

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Example: Moving Against the Field

A:

+3.0 μC

charge is slowly moved from:

100 V

to:

600 V

Potential change:

ΔV = 600 − 100

ΔV = +500 V

Potential-energy change:

ΔEₚ = qΔV

ΔEₚ = (3.0 × 10⁻⁶)(500)

ΔEₚ = +1.5 × 10⁻³ J

Therefore, an external force must supply:

1.5 mJ

of work if there are no other energy transfers.

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Electric Potential Energy in Circuits

Electric potential energy also helps explain energy transfer in circuits.

A battery transfers energy to charges.

When those charges move through circuit components, their electrical energy is transferred into forms such as:

• thermal energy
• light
• kinetic energy
• sound

For example, a potential difference of:

12 V

means:

12 J of energy per coulomb

is transferred between those points.

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Batteries and Energy

A battery does not create charge.

Instead, chemical reactions inside the battery transfer energy to charges and maintain a potential difference.

For example, if:

Q = 5 C

passes through a:

12 V

battery, the energy transferred is:

E = QV

E = 5 × 12

E = 60 J

The chemical energy store of the battery decreases while electrical energy is transferred through the circuit.

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Capacitors and Electric Potential Energy

A capacitor stores energy in its electric field.

The energy stored by a capacitor is:

E = ½CV²

where:

• E = stored energy
• C = capacitance
• V = potential difference

Alternative forms include:

E = ½QV

and:

E = Q²/(2C)

The stored energy can later be transferred when the capacitor discharges.

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Example: Capacitor Energy

A capacitor has:

C = 200 μF

and:

V = 20 V

Convert:

C = 200 × 10⁻⁶ F

Use:

E = ½CV²

E = ½(200 × 10⁻⁶)(20²)

E = 0.040 J

Therefore:

E = 0.040 J

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Energy Transformations in Electric Fields

Electric potential energy can be transformed into many other forms.

Examples include:

Charged particle accelerating

electric potential energy → kinetic energy

Electric motor

electrical energy → kinetic energy + thermal energy

Lamp

electrical energy → light + thermal energy

Heater

electrical energy → thermal energy

Capacitor discharge

stored electric-field energy → electrical energy → other forms

Energy is transferred and transformed, but total energy remains conserved.

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Electric Potential vs Electric Potential Energy

This distinction is essential.

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Potential vs Potential Energy Example

Suppose points A and B have:

V_A = 100 V

V_B = 300 V

The potential difference is:

ΔV = 200 V

This potential difference is the same regardless of what test charge is considered.

However:

For q = +1 C

ΔEₚ = (+1)(200)

ΔEₚ = +200 J

For q = −1 C

ΔEₚ = (−1)(200)

ΔEₚ = −200 J

Same electric field.

Same potential difference.

Different potential-energy change because the charges have different signs.

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Choosing the Correct Equation

Use:

Eₚ = qV

when finding potential energy at a location.

Use:

ΔEₚ = qΔV

when a charge moves between two electric potentials.

Use:

Eₚ = kQq/r

for the potential energy of two point charges.

Use:

W_field = −ΔEₚ

when calculating work done by the electric field.

Use:

Eₚ,i + K_i = Eₚ,f + K_f

when applying conservation of energy.

Use:

K = ½mv²

when connecting energy changes to particle speed.

Use:

E = ½CV²

when calculating energy stored in a capacitor.

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Multi-Step Example

A positive particle has:

q = +4.0 μC

m = 0.002 kg

It starts from rest at:

V_i = 500 V

and moves to:

V_f = 100 V

Find the potential difference

ΔV = 100 − 500

ΔV = −400 V

Find the change in potential energy

ΔEₚ = qΔV

ΔEₚ = (4.0 × 10⁻⁶)(−400)

ΔEₚ = −1.6 × 10⁻³ J

Find the kinetic energy gained

ΔK = −ΔEₚ

ΔK = +1.6 × 10⁻³ J

Since the particle started from rest:

K_f = 1.6 × 10⁻³ J

Find the final speed

K = ½mv²

1.6 × 10⁻³ = ½(0.002)v²

v² = 1.6

v ≈ 1.26 m/s

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Common Mistakes

Mistake: Treating electric potential and electric potential energy as the same quantity.
Potential is measured in volts; potential energy is measured in joules.

Mistake: Forgetting the charge in Eₚ = qV.
Potential energy depends on the charge.

Mistake: Ignoring the sign of a negative charge.
The sign of q is essential when using ΔEₚ = qΔV.

Mistake: Assuming higher electric potential always means higher potential energy.
This is true for positive charges but not necessarily for negative charges.

Mistake: Saying electric potential energy is always positive.
It can be positive, zero, or negative depending on the system and reference.

Mistake: Saying opposite charges have positive potential energy relative to infinity.
Their interaction potential energy is negative.

Mistake: Saying a charge gains kinetic energy without losing another form of energy.
Energy must be conserved.

Mistake: Using Eₚ = kQq/r².
Potential energy varies as 1/r, not 1/r².

Mistake: Assuming moving along an equipotential changes potential energy.
Along an equipotential, ΔV = 0 and therefore ΔEₚ = 0.

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Key Terms

Electric potential energy — energy associated with the positions of charges in an electric field.

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

Potential difference — difference in electric potential between two locations.

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

Joule — SI unit of energy.

Electric field — region in which an electric charge experiences an electrical force.

Work — energy transferred when a force causes displacement.

Conservation of energy — principle that total energy cannot be created or destroyed, only transferred or transformed.

Kinetic energy — energy associated with motion.

Point charge — idealized electric charge concentrated at one location.

Equipotential — line or surface containing points of equal electric potential.

Electron volt — unit of energy equal to 1.602 × 10⁻¹⁹ J.

Reference level — position or configuration chosen to have zero potential energy.

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Key Takeaways

• Electric potential energy is energy associated with the positions of charges in an electric field.
• Electric potential energy is measured in joules.
• Electric potential and electric potential energy are different quantities.
• Electric potential is energy per unit charge.
• Eₚ = qV connects electric potential and potential energy.
• Changes in potential energy are calculated using ΔEₚ = qΔV.
• The sign of the charge must always be considered.
• Positive charges naturally move toward lower electric potential when acted on only by the electric field.
• Negative charges naturally move toward higher electric potential.
• Both movements correspond to decreasing electric potential energy.
• The electric field does positive work when electric potential energy decreases.
• W_field = −ΔEₚ.
• Like charges have positive interaction potential energy relative to infinite separation.
• Opposite charges have negative interaction potential energy relative to infinite separation.
• For two point charges, Eₚ = kQq/r.
• Moving along an equipotential produces no change in electric potential energy.
• If only the electric force acts, ΔK = −ΔEₚ.
• Lost electric potential energy can become kinetic energy.
• Capacitors store energy in electric fields.
• Conservation of energy provides a powerful method for solving electric-field problems.

Check Your Understanding

1. Define electric potential energy.
2. State the SI unit of electric potential energy.
3. Explain the difference between electric potential and electric potential energy.
4. A +3 C charge is at a location with electric potential 20 V. Calculate its electric potential energy.
5. A −2 C charge is at a location with electric potential +50 V. Calculate its electric potential energy.
6. Write the equation relating change in potential energy to charge and potential difference.
7. A +4 C charge moves from 20 V to 70 V. Calculate its change in electric potential energy.
8. Repeat Question 7 for a −4 C charge.
9. Explain why a positive charge naturally tends to move toward lower electric potential.
10. Explain why a negative charge can move toward higher electric potential while still losing potential energy.
11. Describe how the potential energy of two positive charges changes as they are pushed closer together.
12. Describe how the potential energy of opposite charges changes as they move closer together.
13. Write the equation for the electric potential energy of two point charges.
14. Why can electric potential energy be negative?
15. What happens to electric potential energy when a charge moves along an equipotential line?
16. A charge loses 0.50 J of electric potential energy and no other forces do work. How much kinetic energy does it gain?
17. A +5 μC charge moves through a potential drop of 600 V. Calculate its change in electric potential energy.
18. Explain how conservation of energy applies to a charged particle accelerating in an electric field.
19. A particle starts from rest and loses 2.0 × 10⁻³ J of electric potential energy. What is its final kinetic energy?
20. Explain how batteries and capacitors demonstrate the storage and transfer of electrical energy.