Potential Difference Formula

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

The Formula

\Delta V = V_B - V_A = -\int_A^B \vec{E} \cdot d\vec{l}

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

Quick Example

The potential difference across a 9 V battery's terminals is 9 V. Each coulomb of charge gains 9 J of energy passing through the battery.

Notation

\Delta V

is the potential difference in volts (V),

\vec{E}

is the electric field vector,

d\vec{l}

is an infinitesimal displacement along the path, and

q

is the charge in coulombs.

What This Formula Means

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.

Formal View

The potential difference between points

A

and

B

is defined as

\Delta V = V_B - V_A = -\int_A^B \vec{E} \cdot d\vec{l}

, equal to the work done per unit positive charge moved from

A

B

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 \text{ 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

Common Mistakes

Confusing electric potential (at one point) with potential difference (between two points) — voltage is always a difference. - Fix this by naming the system, checking "Am I using a field or potential to explain how one object influences another across space?", and attaching units or direction to the final statement.
Forgetting that potential difference is measured in volts (joules per coulomb), not in joules alone. - Fix this by naming the system, checking "Am I using a field or potential to explain how one object influences another across space?", and attaching units or direction to the final statement.
Mixing up the sign convention: work done by the field on a positive charge moving from high to low potential is positive. - Fix this by naming the system, checking "Am I using a field or potential to explain how one object influences another across space?", and attaching units or direction to the final statement.
Using potential difference from a keyword alone - Signal words like field, charge, magnet only point to a possible model; the system must match too.

Why This Formula Matters

Potential Difference gives students a way to explain non-contact forces and energy changes. It connects electricity, magnetism, gravitation, induction, motors, generators, and orbital motion through a shared spatial model.

Frequently Asked Questions

What is the Potential Difference formula?

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

How do you use the Potential Difference formula?

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

What do the symbols mean in the Potential Difference formula?

$\Delta V$ is the potential difference in volts (V), $\vec{E}$ is the electric field vector, $d\vec{l}$ is an infinitesimal displacement along the path, and $q$ is the charge in coulombs.

Why is the Potential Difference formula important in Physics?

What do students get wrong about Potential Difference?

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.

What should I learn before the Potential Difference formula?

Before studying the Potential Difference formula, you should understand: electric potential, voltage.

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Electric Potential Voltage

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Electric Potential Formula Voltage Formula