1. Concept Overview
Electric potential difference tells us how much extra energy per unit charge is gained or lost when moving between two points in an electric field. It is similar to a height difference in gravitational energy: a charge naturally moves from higher potential to lower potential. This difference is what provides the “push” that drives charge flow in circuits and natural electric processes.
2. Definition
Electric potential difference between two points A and B is the work done by the electric field in moving a unit positive test charge from A to B.
Mathematically:
\( V_B - V_A = \dfrac{W_{AB}}{q} \)
3. Meaning of Potential Difference
Potential difference measures how strongly the electric field pushes a charge from one point to another. A larger difference means a stronger push. If a positive charge is released, it naturally moves from higher potential to lower potential because it loses potential energy along the way.
3.1. Energy Interpretation
If a charge \( q \) moves from A to B, the change in potential energy is:
\( \Delta U = q (V_B - V_A) \)
This connects potential difference to the actual energy change of a charge.
4. Relation Between Electric Field and Potential Difference
Electric potential difference is directly linked to how electric field varies along a path.
4.1. Formula
In one dimension:
\( V_B - V_A = -\int_A^B \vec{E} \cdot d\vec{l} \)
For a uniform field:
\( V_B - V_A = -E d \)
Here, \( d \) is the displacement along the direction of the field.
4.2. Interpretation
A strong electric field means the potential changes rapidly over a short distance. A weak electric field changes the potential gradually.
5. Potential Difference Is Independent of Path
Electric potential is a scalar quantity, and for electrostatic fields, potential difference between two points depends only on the initial and final points, not the path taken to get from one to the other.
5.1. Reason
Electrostatic fields are conservative. This means the work done by the field around any closed path is zero.
\( \oint \vec{E} \cdot d\vec{l} = 0 \)
6. Worked Examples
6.1. Example 1: Uniform Electric Field
In a uniform electric field of \( 200 \text{ N/C} \), two points are \( 0.15 \text{ m} \) apart along the direction of the field. The potential difference is:
\( V_B - V_A = -Ed = -200 \times 0.15 = -30 \text{ V} \).
This means B is at a lower potential than A.
6.2. Example 2: Moving Opposite the Field
If the same 200 N/C field is crossed in the opposite direction, the potential difference becomes:
\( V_B - V_A = +30 \text{ V} \).
So B is now at a higher potential.
7. Physical Interpretation
Electric potential difference is similar to the height difference between two points in gravity. A charge naturally “rolls” from higher potential to lower potential, gaining kinetic energy or releasing energy in other forms. This difference in potential is what drives current in circuits and explains the natural movement of charges in electric fields.