Potential Difference Formula

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 to 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?

Solution

  1. 1
    Potential difference (voltage) is defined as energy per unit charge: V = \frac{W}{Q}.
  2. 2
    For 1 \text{ C} of charge: W = VQ = 9 \times 1 = 9 \text{ J}.
  3. 3
    Each coulomb of charge gains 9 \text{ J} of electrical energy from the battery.

Answer

W = 9 \text{ J per coulomb}
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
A 12 \text{ V} battery drives 3 \text{ A} of current through a resistor. Find the resistance and the power dissipated.

Common Mistakes

  • Confusing electric potential (at one point) with potential difference (between two points) — voltage is always a difference.
  • Forgetting that potential difference is measured in volts (joules per coulomb), not in joules alone.
  • Mixing up the sign convention: work done by the field on a positive charge moving from high to low potential is positive.

Why This Formula Matters

Links the abstract concept of electric field to the practical concept of voltage used in circuits and power systems.

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?

Links the abstract concept of electric field to the practical concept of voltage used in circuits and power systems.

What do students get wrong about Potential Difference?

Voltage is always measured between two points — saying 'the voltage at this wire' implicitly means relative to ground.

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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Related Concepts

Electric PotentialVoltage