Electric potential difference

Electric Potential Difference – Concept and Work-Energy Relation

In electrostatics, the electric potential difference between two points describes how much work is done in moving a charge between those points in an electric field. This concept is crucial for understanding energy transfer in electric fields and circuits.

Definition of Electric Potential Difference

The electric potential difference \( \Delta V \) between two points A and B is defined as the work done per unit charge by an external force to move a test charge from point A to point B without acceleration.

Mathematically:

\( \Delta V = \frac{\Delta W}{q} \)

or rearranged:

\( \Delta W = q \Delta V \)

• \( \Delta W \) – Work done to move the charge (in joules)
• \( q \) – Magnitude of the charge (in coulombs)
• \( \Delta V = V_B - V_A \) – Electric potential difference between points B and A (in volts)

Understanding How \( \Delta W = q \Delta V \) Comes

Electric potential at any location refers to the amount of potential energy associated with a unit positive charge placed at that point. When a charge is transferred from point A with potential \( V_A \) to point B with potential \( V_B \), the corresponding change in its potential energy can be expressed as:

\( \Delta U = U_B - U_A \)

The work done by an external force to move the charge without acceleration is equal to this change in potential energy:

\( \Delta W = \Delta U = q(V_B - V_A) = q \Delta V \)

Differential Form: \( dW = \vec{F} \cdot d\vec{x} \)

To understand this relation more precisely, consider moving a small charge \( q \) through a very small displacement \( d\vec{x} \) in an electric field \( \vec{E} \). The infinitesimal work done is:

\( dW = \vec{F} \cdot d\vec{x} = q \vec{E} \cdot d\vec{x} \)

From this, we get:

\( dV = \frac{dW}{q} = \vec{E} \cdot d\vec{x} \)

or equivalently,

\( dW = q\, dV \)

Electrostatic Analogy of Potential Difference

To better understand this, we can use an analogy with gravity. Just as moving a mass in a gravitational field involves work and change in potential energy, moving a charge in an electric field involves similar energy changes.

Gravitational analogy:

• Gravitational potential difference: \( \Delta V_g = \frac{\Delta W_g}{m} \)
• Electric potential difference: \( \Delta V = \frac{\Delta W}{q} \)

This helps students visualize electric potential as a kind of "height" in an energy landscape.

Key Concepts and Takeaways

• Electric potential difference is path-independent in electrostatics (conservative field).
• To move a charge from one point to another with a different electric potential, external work must be done, given by \( \Delta W = q \Delta V \).
• If \( \Delta V > 0 \), positive work is done by an external agent (against the electric field).
• SI unit of potential difference is the volt (V), where \( 1\, {V} = 1\, {J}/{C} \).