Potential Difference

What Is Potential Difference?

Potential difference is the amount of energy transferred, or work done, per unit charge as charge moves between two points in an electric circuit or electric field.

Potential difference is commonly called:

voltage

It is represented by:

V

The basic relationship is:

V = W/Q

where:

• V = potential difference (V)
• W = work done or energy transferred (J)
• Q = charge transferred (C)

This means that voltage tells us how much energy is transferred for each coulomb of charge.

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The Volt

The SI unit of potential difference is the:

volt (V)

From:

V = W/Q

we obtain:

1 V = 1 J/C

Therefore:

1 volt means 1 joule of energy transferred per coulomb of charge.

For example, if a component has a potential difference of:

6 V

then each coulomb of charge transfers:

6 J

of energy as it passes through the component.

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Interpreting Voltage

Voltage is sometimes described informally as the electrical "push" that drives current.

This can be useful, but a more precise description is:

potential difference represents an energy difference per unit charge.

A potential difference creates an electric field within a conductor.

That electric field exerts forces on charged particles and causes a net movement of charge.

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Potential Difference and Electric Potential

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

Potential difference compares the electric potentials at two locations.

For points A and B:

ΔV = V_B − V_A

Therefore, potential difference tells us how electric potential changes between two points.

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A Gravitational Analogy

Potential difference can be compared loosely with a difference in height.

Water at a higher gravitational position can move downward and transfer energy.

Similarly, charges moving through an electric potential difference can transfer energy.

However, voltage is not literally a height or pressure.

It is:

energy transferred per unit charge

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Voltage and Charge Movement

When a potential difference exists across a conductor, an electric field is established within the conductor.

This electric field exerts forces on mobile charges.

As a result:

potential difference → electric field → charge movement

If a complete conducting path exists, this can produce an electric current.

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Conventional Current

By convention, current direction is defined as the direction positive charge would move.

Therefore:

conventional current flows from higher potential toward lower potential through passive circuit components

In metallic wires, however, the mobile charge carriers are electrons.

Electrons move in the opposite direction to conventional current.

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Why Electrons Move the Opposite Way

Electrons have negative charge.

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

Because electrons are negatively charged, the electrical force on them acts opposite to the electric field.

Therefore:

conventional current direction ≠ electron drift direction

in a metal conductor.

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Batteries and Potential Difference

A battery maintains a potential difference between its terminals.

Chemical reactions inside the battery transfer energy to charges.

For example, a:

9 V battery

provides approximately:

9 J of energy per coulomb

of charge moved through the battery under idealized conditions.

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Energy Gain in a Battery

When charge moves through a battery from lower electric potential to higher electric potential, it gains electric potential energy.

For:

Q = 1 C

and:

V = 12 V

the energy transferred is:

W = QV

W = 1 × 12

W = 12 J

The battery has transferred 12 J of energy to the charge.

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Energy Transfer in a Component

When charge passes through a component such as a resistor or lamp, electrical energy can be transferred into other forms.

For example:

Lamp

electrical energy → light + thermal energy

Heater

electrical energy → thermal energy

Motor

electrical energy → kinetic energy + thermal energy + sound

Speaker

electrical energy → sound + thermal energy

The potential difference across the component describes the energy transferred per coulomb.

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Rearranging the Voltage Equation

Starting with:

V = W/Q

we can rearrange it.

For energy or work:

W = QV

For charge:

Q = W/V

These three forms are useful for solving different types of problems.

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Example: Calculating Potential Difference

Suppose:

W = 24 J

Q = 4 C

Use:

V = W/Q

V = 24/4

V = 6 V

Therefore, 6 J of energy is transferred for every coulomb of charge.

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Example: Calculating Energy

A charge of:

Q = 5 C

passes through a potential difference of:

V = 12 V

Use:

W = QV

W = 5 × 12

W = 60 J

Therefore:

60 J of energy is transferred.

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Example: Calculating Charge

A device transfers:

W = 180 J

when operating across:

V = 12 V

Use:

Q = W/V

Q = 180/12

Q = 15 C

Therefore:

15 C of charge passed through the device.

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Charge and Current

Charge transferred is related to current by:

Q = It

where:

• Q = charge (C)
• I = current (A)
• t = time (s)

This equation can be combined with:

W = QV

to give:

W = VIt

This allows energy transfer to be calculated from:

• voltage
• current
• time

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Example: Voltage, Current and Energy

A heater operates at:

V = 12 V

with current:

I = 3 A

for:

t = 20 s

First find charge:

Q = It

Q = 3 × 20

Q = 60 C

Then:

W = QV

W = 60 × 12

W = 720 J

Therefore:

720 J of electrical energy is transferred.

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Direct Calculation Using W = VIt

The previous problem can be solved directly:

W = VIt

W = 12 × 3 × 20

W = 720 J

Both methods give the same result.

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Potential Difference and Power

Power is the rate of energy transfer:

P = W/t

Since:

W = VIt

then:

P = VIt/t

Therefore:

P = VI

This is one of the most important equations in electricity.

Voltage therefore connects directly to both:

energy transfer

and:

power

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Example: Electrical Power

A motor operates at:

V = 24 V

and draws:

I = 2.5 A

Power:

P = VI

P = 24 × 2.5

P = 60 W

The motor receives electrical energy at a rate of:

60 J/s

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Potential Difference and Resistance

For an ohmic conductor:

V = IR

where:

• V = potential difference
• I = current
• R = resistance

Therefore:

I = V/R

For a fixed resistance, increasing the potential difference increases the current.

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Example: Voltage and Current

A resistor has:

R = 20 Ω

and potential difference:

V = 10 V

Current:

I = V/R

I = 10/20

I = 0.50 A

If the potential difference doubles to:

20 V

then:

I = 20/20

I = 1.0 A

For the same resistance, doubling voltage doubles current.

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Potential Difference Across a Resistor

When charge moves through a resistor, electrical energy is transferred mainly into thermal energy.

If the potential difference is:

10 V

then every coulomb passing through the resistor transfers:

10 J

of electrical energy.

This energy increases the internal energy of the resistor and its surroundings.

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Potential Rise and Potential Drop

When moving around a circuit, we can describe changes in potential.

Through a battery

There is a:

potential rise

because energy is transferred to the charges.

Through a resistor

There is a:

potential drop

because energy is transferred from the electrical system to thermal energy.

Through a motor

There is a potential drop because electrical energy is transferred into kinetic energy and other forms.

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Potential Difference Around a Simple Circuit

Consider a:

12 V battery

connected to one resistor.

Ideally, the resistor has:

12 V

across it.

The battery provides:

12 J/C

and the resistor transfers:

12 J/C

into other forms.

Energy is conserved.

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Voltage in a Series Circuit

In a series circuit, the supply potential difference is shared between the components.

For example:

V_supply = 12 V

V₁ = 4 V

V₂ = 8 V

Then:

V_supply = V₁ + V₂

12 = 4 + 8

This reflects conservation of energy.

Each coulomb gains 12 J from the source and then transfers:

4 J + 8 J = 12 J

through the components.

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Voltage in Parallel Circuits

Components connected in parallel have the same potential difference across them.

If three lamps are connected directly in parallel across a:

12 V battery

then each branch has approximately:

12 V

across it.

Each coulomb passing through any branch can therefore transfer:

12 J

through that branch.

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Measuring Potential Difference

Potential difference is measured using a:

voltmeter

A voltmeter is connected:

in parallel

with the component being measured.

This allows the meter to compare the electric potentials on opposite sides of the component.

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Why a Voltmeter Is Connected in Parallel

Potential difference is a comparison between two locations.

Therefore, the voltmeter must connect to:

both sides of the component

A good voltmeter has very high resistance so that it draws very little current and causes minimal disturbance to the circuit.

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Voltmeter vs Ammeter

A voltmeter measures:

potential difference

and is connected:

in parallel

An ammeter measures:

current

and is connected:

in series

Confusing these connections is a common circuit error.

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Potential Difference in an Electric Field

Potential difference also exists between locations in an electric field.

For a uniform electric field:

|ΔV| = Ed

where:

• E = electric field strength (V/m)
• d = distance along the field direction (m)

Therefore:

E = |ΔV|/d

A stronger field produces a larger change in potential over the same distance.

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Example: Uniform Electric Field

Two parallel plates are:

0.020 m

apart.

The electric field strength is:

5000 V/m

Potential difference:

|ΔV| = Ed

|ΔV| = 5000 × 0.020

|ΔV| = 100 V

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Work Done in an Electric Field

Suppose a charge:

Q = 2.0 × 10⁻⁶ C

moves through a potential difference of:

500 V

Energy transferred:

W = QV

W = (2.0 × 10⁻⁶)(500)

W = 1.0 × 10⁻³ J

or:

W = 1.0 mJ

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Example with Milliamps

A device operates at:

V = 9 V

Current:

I = 250 mA

for:

t = 60 s

Convert:

250 mA = 0.250 A

Energy:

W = VIt

W = 9 × 0.250 × 60

W = 135 J

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Example with Microcoulombs

A charge of:

250 μC

moves through:

400 V

Convert:

250 μC = 250 × 10⁻⁶ C

Then:

W = QV

W = (250 × 10⁻⁶)(400)

W = 0.100 J

Unit conversion is especially important in electrical calculations.

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Choosing the Correct Equation

Use:

V = W/Q

when given energy and charge.

Use:

W = QV

when calculating energy transfer.

Use:

Q = W/V

when finding charge.

Use:

Q = It

when current and time are given.

Use:

W = VIt

when voltage, current, and time are known.

Use:

P = VI

when calculating electrical power.

Use:

V = IR

for voltage, current, and resistance in an ohmic component.

Use:

|ΔV| = Ed

for a uniform electric field.

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Multi-Step Example

A 6 V motor draws:

2 A

for:

5 minutes.

Calculate the electrical energy transferred.

First convert time:

5 min = 300 s

Use:

W = VIt

W = 6 × 2 × 300

W = 3600 J

or:

W = 3.6 kJ

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Alternative Solution

First calculate charge:

Q = It

Q = 2 × 300

Q = 600 C

Then:

W = QV

W = 600 × 6

W = 3600 J

Again:

W = 3.6 kJ

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Comparing Two Devices

Device A operates across:

6 V

Device B operates across:

12 V

Each coulomb passing through Device A transfers:

6 J

Each coulomb passing through Device B transfers:

12 J

Therefore, Device B transfers twice as much energy per coulomb.

This does not automatically mean Device B uses twice as much total energy because total energy also depends on the amount of charge transferred.

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Voltage Does Not Equal Energy

Voltage is not measured in joules.

Voltage is:

energy per unit charge

For example:

12 V = 12 J/C

To calculate actual energy transferred, we also need to know how much charge moves:

W = QV

This distinction is essential.

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Voltage Does Not Equal Current

Voltage and current are different quantities.

Voltage

Measures:

energy transferred per unit charge

Unit:

volt (V)

Current

Measures:

rate of flow of charge

I = Q/t

Unit:

ampere (A)

Voltage can cause current to flow when there is a suitable conducting path, but voltage and current are not the same thing.

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What Determines Current?

The current depends on both:

• potential difference
• resistance

For an ohmic conductor:

I = V/R

Therefore:

larger V → larger current if R is constant

larger R → smaller current if V is constant

A large voltage does not automatically mean a large current if resistance is also very large.

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Potential Difference and Energy Conservation

Potential difference provides an important way to understand energy conservation in circuits.

Consider a circuit with:

12 V source

and two components having drops of:

5 V and 7 V

Each coulomb gains:

12 J

from the source.

Then transfers:

5 J + 7 J = 12 J

through the components.

Therefore:

energy gained = energy transferred

This is the energy basis of Kirchhoff's Voltage Law.

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Kirchhoff's Voltage Law

Around a complete closed loop:

total potential rises = total potential drops

or equivalently:

ΣV = 0

This follows from conservation of energy.

A charge returning to its original point in a circuit must return to the same electric potential.

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Potential Difference and Capacitors

A capacitor stores energy when a potential difference exists across its plates.

For a capacitor:

Q = CV

Therefore:

V = Q/C

As charge accumulates on the capacitor, the potential difference across it increases.

The stored energy is:

E = ½CV²

This is another example of voltage being closely connected to electrical energy.

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Everyday Voltage Examples

Different electrical systems operate at very different voltages.

Examples include:

• small batteries
• USB-powered electronics
• vehicle electrical systems
• household electricity
• transmission systems
• electronic circuits

The appropriate voltage depends on the purpose and design of the system.

Higher voltage can allow greater energy transfer per coulomb, but it can also create greater electrical hazards.

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Why High Voltage Is Used in Power Transmission

Electrical power is:

P = VI

For the same transmitted power, increasing voltage allows current to decrease.

Power lost as heating in transmission wires is:

P_loss = I²R

Therefore, reducing current greatly reduces transmission losses.

This is why electrical power is transmitted over long distances at high potential differences.

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Potential Difference and Electrical Safety

Potential difference can cause current to flow through the human body if a conducting path exists.

The current depends on factors including:

• voltage
• body resistance
• contact conditions
• current path
• duration

Therefore, voltage alone does not completely determine electrical danger.

Electrical safety requires preventing dangerous current from passing through the body.

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Common Mistakes

Mistake: Saying voltage is the amount of charge flowing.
Voltage is energy transferred per unit charge.

Mistake: Saying 1 V = 1 J.
The correct relationship is 1 V = 1 J/C.

Mistake: Confusing current with potential difference.
Current is charge flow rate; voltage is energy per charge.

Mistake: Using W = V/Q.
The correct rearrangement is W = QV.

Mistake: Forgetting to convert mA to A or μC to C.

Mistake: Saying a battery creates charge.
A battery transfers energy to charges and maintains a potential difference.

Mistake: Connecting a voltmeter in series.
A voltmeter is connected in parallel.

Mistake: Assuming higher voltage always means greater total energy use.
Total energy also depends on the charge transferred.

Mistake: Saying electrons move in the same direction as conventional current in metals.
Electron drift is opposite to conventional current.

Mistake: Ignoring energy conservation when analyzing voltage around a circuit.
Total potential rises and drops around a closed loop must balance.

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Key Terms

Potential difference — work done or energy transferred per unit charge between two points.

Voltage — common name for potential difference.

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

Charge (Q) — quantity of electric charge, measured in coulombs.

Coulomb (C) — SI unit of electric charge.

Work (W) — energy transferred when a force causes displacement or when charge moves through a potential difference.

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

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

Current — rate of flow of electric charge.

Resistance — opposition to current flow.

Potential rise — increase in electric potential, such as across a source.

Potential drop — decrease in electric potential across a component transferring electrical energy.

Voltmeter — instrument used to measure potential difference.

Conventional current — current direction defined as the direction positive charge would move.

Electrical power — rate of electrical energy transfer.

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Key Takeaways

• Potential difference is work done or energy transferred per unit charge.
• Potential difference is commonly called voltage.
• V = W/Q.
• 1 V = 1 J/C.
• A potential difference establishes an electric field that can cause charges to move.
• Voltage and current are different quantities.
• Current measures charge flow per unit time.
• Voltage measures energy transfer per unit charge.
• Batteries maintain potential differences by transferring energy to charges.
• Components transfer electrical energy into other forms.
• W = QV can be used to calculate electrical energy transfer.
• Charge can be calculated using Q = It.
• Combining these gives W = VIt.
• Electrical power is P = VI.
• For an ohmic conductor, V = IR.
• A voltmeter measures potential difference and is connected in parallel.
• In series circuits, potential differences across components add to the supply voltage.
• Parallel branches have the same potential difference across them.
• In a uniform electric field, |ΔV| = Ed.
• Potential rises and drops around a complete circuit reflect conservation of energy.
• Understanding voltage as energy per coulomb provides a powerful way to analyze electrical circuits.

Check Your Understanding

1. Define potential difference.
2. What does a potential difference of 1 V mean?
3. Explain the difference between voltage and current.
4. Calculate the potential difference if 30 J of energy is transferred by 5 C of charge.
5. Calculate the energy transferred when 8 C moves through 12 V.
6. A device transfers 240 J of energy through a potential difference of 20 V. Calculate the charge transferred.
7. Explain how a potential difference can cause charges to move.
8. What energy transfer occurs when charge passes through a resistor?
9. A 9 V battery transfers 45 J of energy. Calculate the charge moved through the battery.
10. A current of 2 A flows for 30 s. Calculate the charge transferred.
11. A 12 V device draws 3 A for 20 s. Calculate the energy transferred.
12. A motor operates at 24 V and draws 4 A. Calculate its electrical power.
13. Explain why a voltmeter is connected in parallel.
14. Two resistors in series have potential differences of 3 V and 7 V. Calculate the supply voltage.
15. Three lamps are connected in parallel across a 12 V battery. What potential difference is across each lamp?
16. A uniform electric field has strength 4000 V/m. Calculate the potential difference across 0.050 m along the field.
17. Explain the difference between a potential rise and a potential drop.
18. Explain how voltage demonstrates conservation of energy around a complete circuit.
19. Explain why high voltage is useful for long-distance electrical power transmission.
20. A 230 V appliance draws 2.0 A for 5 minutes. Calculate the charge transferred and the electrical energy transferred.