1. Concept Overview
Electric potential gives a simple way to talk about electric energy. Instead of focusing on forces directly, potential tells us how much energy a unit positive charge would have at a point in the electric field. Just like height determines gravitational potential energy, electric potential determines electric potential energy.
2. Definition
Electric potential at a point is defined as the work done per unit positive test charge in bringing it from infinity to that point, without acceleration.
Mathematically:
\( V = \dfrac{W}{q} \)
3. Electric Potential and Potential Energy
If a charge \( q \) is placed at a point where the potential is \( V \), then the potential energy is:
\( U = qV \)
This connects the ideas of potential and energy in a simple way.
3.1. Interpretation
A point with high potential means a positive test charge has high potential energy there. A negative potential means the charge would naturally move toward that region.
4. Potential Difference
The difference in potential between two points measures how much work the electric field does when moving a unit charge between them. It is the "push" that drives charges in circuits and natural electric processes.
4.1. Formula
Potential difference between two points A and B:
\( V_B - V_A = \dfrac{W_{AB}}{q} \)
If positive work is required to move the charge, the potential increases.
5. Electric Potential Due to a Point Charge
A point charge creates a potential around it that depends only on the distance from the charge.
5.1. Formula
For a point charge \( Q \), potential at distance \( r \) is:
\( V = \dfrac{1}{4\pi\varepsilon_0} \dfrac{Q}{r} \)
Positive for a positive charge and negative for a negative charge.
5.2. Notes
- Potential falls as distance increases.
- Potential is a scalar, so potentials from multiple charges add directly.
6. Relation Between Electric Field and Potential
Electric potential changes whenever we move in the direction of the electric field. The rate of change relates the two quantities.
6.1. Formula
The electric field is the negative gradient of potential:
\( \vec{E} = -\nabla V \)
In one dimension:
\( E = -\dfrac{dV}{dx} \)
6.2. Interpretation
A large change in potential over a short distance means a strong electric field, while small change means a weak field.
7. Characteristics of Electric Potential
- It is a scalar quantity.
- It can be positive or negative depending on the charge creating it.
- Potential at infinity is taken as zero.
- Potential from multiple charges adds directly.
8. Worked Examples
8.1. Example 1: Potential at a Point Due to a Charge
A charge \( Q = 6 \times 10^{-6} \text{ C} \) is placed at a distance of \( 0.2 \text{ m} \). The potential is:
\( V = \dfrac{1}{4\pi\varepsilon_0} \dfrac{Q}{r} = 9 \times 10^9 \times \dfrac{6 \times 10^{-6}}{0.2} = 2.7 \times 10^5 \text{ V} \).
8.2. Example 2: Potential Difference
If the potential at A is \( 100 \text{ V} \) and at B is \( 40 \text{ V} \), then the potential difference is:
\( V_B - V_A = 40 - 100 = -60 \text{ V} \).
This indicates that the electric field would naturally push a positive charge from A toward B.
9. Physical Interpretation
You can think of electric potential like electric "height". A positive charge naturally moves from higher potential to lower potential, similar to how an object rolls downhill in gravity. This viewpoint helps explain movement of charges and the behaviour of electric fields more intuitively.