Potential Difference Examples: 47 Problems with Answers

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Potential Difference.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Physics.

Concept Recap

The difference in electric potential between two points, equal to the work done per unit charge moving between them.

Potential difference is the 'height drop' that makes charges flow — the bigger the drop, the harder the push.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Potential Difference starts by naming the source, the object affected, and how the field or potential changes through space.

Common stuck point: Students often know a formula related to potential difference but skip the recognition step: Am I using a field or potential to explain how one object influences another across space? That leads to a correct-looking substitution attached to the wrong physical model.

Sense of Study hint: Ask: Am I using a field or potential to explain how one object influences another across space?

Worked Examples

Example 1

easy

A battery has a potential difference of

9 \text{ V}

across its terminals. How much energy does it give to each coulomb of charge that passes through it?

Answer

W = 9 J per coulomb

First step

Potential difference (voltage) is defined as energy per unit charge:

V = \frac{W}{Q}

Full solution

2
For $1 \text{ C}$ of charge: $W = VQ = 9 \times 1 = 9 \text{ J}$ .
3
Each coulomb of charge gains $9 \text{ J}$ of electrical energy from the battery.

Potential difference measures the energy transferred per unit charge between two points. A

9 \text{ V}

battery gives

9 \text{ J}

of energy to every coulomb of charge, which is then dissipated in the circuit components.

Example 2

medium

A current of

0.5 \text{ A}

flows through a

10 \text{ } \Omega

resistor. What is the potential difference across the resistor? How much energy is dissipated in

20 \text{ s}

Example 3

medium

12 \text{ V}

battery drives

3 \text{ A}

of current through a resistor. Find the resistance and the power dissipated.

Find R when 12 V drives 3 A

Example 4

medium

Two parallel plates are separated by

5 \text{ mm}

and have a potential difference of

250 \text{ V}

. Find the magnitude of the electric field between them.

Example 5

medium

Two parallel resistors of

3 \text{ }\Omega

and

6 \text{ }\Omega

are connected across a

12 \text{ V}

source. What is the potential difference across each resistor?

Parallel resistors across a 12 V source

Example 6

medium

In a circuit,

V_A = -3 \text{ V}

and

V_B = +7 \text{ V}

. (a) What is

V_B - V_A

? (b) If a

+0.5 \text{ C}

charge moves from B to A, what is the work done by the electric field?

Example 7

hard

A capacitor with

C = 100 \text{ }\mu\text{F}

is charged through a

1 \text{ k}\Omega

resistor by a

10 \text{ V}

source. What is the voltage across the capacitor after one time constant?

Example 8

hard

Two identical capacitors of

C = 50 \text{ }\mu\text{F}

are connected in series across a

24 \text{ V}

source. Find the voltage across each capacitor and the total charge stored.

Example 9

challenge

A spherical conductor of radius

R = 0.10 \text{ m}

carries total charge

Q = +20 \text{ nC}

. Find the potential difference between its surface and a point at

r = 0.30 \text{ m}

from the center. Use

k = 9.0 \times 10^9 \text{ N m}^2/\text{C}^2

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

medium

Two points A and B in an electric field have potentials

V_A = 100 \text{ V}

and

V_B = 40 \text{ V}

. What is the potential difference from A to B? How much work is done moving a

2 \times 10^{-6} \text{ C}

charge from B to A?

Example 2

hard

In a circuit with a

12 \text{ V}

battery, three resistors in series have values

2 \text{ } \Omega

4 \text{ } \Omega

, and

6 \text{ } \Omega

. What is the potential difference across each? Verify that they sum to the battery voltage.

Series resistors across a 12 V battery

Example 3

easy

Moving

4

C through a region does

20

J of work. Find the potential difference.

Example 4

easy

What are the units of potential difference?

Example 5

easy

3

C charge crosses a

6

V difference. Find the work done.

Example 6

easy

Is potential difference a value at one point or between two points?

Example 7

easy

Points A and B have potentials

12

V and

5

V. Find

V_A - V_B

Example 8

easy

A uniform field

E=200

N/C spans a gap

d=0.5

m. Find the potential difference.

Example 9

easy

9

V battery moves

2

C. Find the energy delivered.

Example 10

easy

Voltage is sometimes called what everyday quantity that 'pushes' current?

Example 11

medium

A field

E=400

N/C spans

d=0.25

m. A

2

C charge crosses it. Find the work done on the charge.

Example 12

medium

An electron is accelerated through

100

V. Find the kinetic energy gained in joules. (

e=1.6\times10^{-19}

C.)

Example 13

medium

To move

5

C from A to B requires

40

J against the field. Find

V_B - V_A

Example 14

medium

A uniform field gives

\Delta V = 60

V across

d=0.2

m. Find the field strength.

Example 15

medium

Crossing a

50

V difference, a charge gains

200

J. Find the charge.

Example 16

medium

+2

C charge moves from a point at

3

V to a point at

8

V. Find the work done by the field on it.

Example 17

medium

A capacitor gap has

E=2000

N/C. The plates are

0.01

m apart. A

3

C charge crosses fully. Find the work.

Example 18

challenge

A proton (

q=1.6\times10^{-19}

m=1.67\times10^{-27}

kg) starts at rest and is accelerated through

200

V. Find its final speed.

Example 19

challenge

In a uniform field, point A is at

20

V and B at

5

V, separated by

0.3

m along the field. Find both the field strength and the work to move

+2

C from A to B.

Example 20

challenge

Two charges

+q

and

+q

are fixed

2d

apart. Find the potential difference between the midpoint and a point at distance

d

beyond one charge along the line. Express in terms of

k,q,d

Example 21

medium

4

C charge crosses a region and the field does

48

J of work on it. Find the potential difference (drop).

Example 22

medium

A uniform field of

250

N/C spans

0.4

m. A

3

C charge crosses fully. Find the work done.

Example 23

easy

A charge of

0.2 \text{ C}

gains

3 \text{ J}

of energy as it moves between two points. What is the potential difference between the points?

Example 24

easy

A uniform field has

E = 500 \text{ V/m}

pointing from plate A to plate B, with separation

d = 0.04 \text{ m}

. Find

|V_A - V_B|

Example 25

easy

Moving a

+2 \text{ C}

charge from A to B against the field requires

24 \text{ J}

of external work. What is

V_B - V_A

Example 26

medium

An electron (

q = -1.6 \times 10^{-19} \text{ C}

) is accelerated from rest through a potential difference of

200 \text{ V}

. Find its final kinetic energy in joules and in electron-volts.

Example 27

medium

1.5 \text{ V}

AA battery delivers

0.30 \text{ A}

for

30 \text{ minutes}

. How much energy does the battery supply?

Example 28

medium

A point charge

Q = +5 \text{ nC}

sits at the origin. Find the potential difference

V(0.10 \text{ m}) - V(0.50 \text{ m})

along a radial line. Use

k = 9.0 \times 10^9 \text{ N m}^2/\text{C}^2

Example 29

medium

A capacitor of

C = 220 \text{ }\mu\text{F}

is charged to

9 \text{ V}

. Find the stored charge and the energy stored.

Example 30

medium

A wire carries

2.0 \text{ A}

and dissipates

24 \text{ W}

. What is the potential difference across the wire?

Example 31

medium

6 \text{ V}

battery is connected across the series combination of a

2 \text{ }\Omega

resistor and a

4 \text{ }\Omega

resistor. Find the current and the voltage across the

4 \text{ }\Omega

resistor.

Series circuit — find I and V across 4 Ω

Example 32

hard

A proton (

m = 1.67 \times 10^{-27} \text{ kg}

q = 1.6 \times 10^{-19} \text{ C}

) starts from rest and is accelerated through a potential difference of

1000 \text{ V}

. Find its final speed.

Example 33

hard

Two point charges

+4 \text{ nC}

and

-2 \text{ nC}

sit at

x = 0

and

x = 0.20 \text{ m}

respectively. Find the electric potential at

x = 0.10 \text{ m}

. Use

k = 9.0 \times 10^9 \text{ N m}^2/\text{C}^2

Example 34

hard

12 \text{ V}

battery with internal resistance

0.5 \text{ }\Omega

delivers current to an external

5.5 \text{ }\Omega

load. Find the terminal voltage across the battery.

Battery with internal resistance — find terminal voltage

Example 35

hard

A uniform field

E = 800 \text{ V/m}

points in the

+x

direction. Find

V_B - V_A

if A is at

(0, 0)

and B is at

(0.05 \text{ m}, 0.10 \text{ m})

Example 36

hard

Going around a single loop circuit, you measure voltage drops of

4 \text{ V}

3 \text{ V}

, and

5 \text{ V}

across three resistors. What must the EMF of the source in the loop be (assume ideal)?

Example 37

hard

A point at distance

r_1 = 0.50 \text{ m}

from a charge

Q

has potential

V_1 = 36 \text{ V}

. Find the potential at

r_2 = 0.30 \text{ m}

from the same charge.

Example 38

challenge

An alpha particle (

q = 2e = 3.2 \times 10^{-19} \text{ C}

m = 6.64 \times 10^{-27} \text{ kg}

) is accelerated from rest through

5000 \text{ V}

. Find its final speed.

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Related Concepts

Electric Potential Voltage

Background Knowledge

These ideas may be useful before you work through the harder examples.

electric potentialvoltage