19.3 Electrical Potential Due to a Point Charge

Electric Potential of a Point Charge

Electric potential describes the energy per unit charge at a point in an electric field. For a point charge, the potential decreases with distance, following an inverse relationship. Understanding electric potential helps explain how charges interact and move in electric fields.

A crucial distinction: while the electric field is a vector (it has direction), electric potential is a scalar quantity measured in volts. This makes potential easier to work with in many calculations since you don't need to worry about components or directions.

Equation for Electric Potential

The electric potential ($V$) at a distance $r$ from a point charge $q$ is given by:

$V = \frac{kq}{r}$

• $k$ is Coulomb's constant: $8.99 \times 10^9 \, \frac{N \cdot m^2}{C^2}$
• $q$ is the point charge in coulombs (C)
• $r$ is the distance from the charge to the point of interest in meters (m)

Electric potential is measured in volts (V), which are equivalent to joules per coulomb (J/C). Because potential is a scalar, you only need the magnitude and sign of the charge, not any direction.

A few things to notice about this equation:

• Potential is inversely proportional to distance ($1/r$, not $1/r^2$). Double the distance and the potential drops to half.
• A positive charge creates positive potential around it; a negative charge creates negative potential.
• Points at equal distances from the charge all have the same potential. These form equipotential surfaces, which are spherical shells centered on the point charge.

Electric Potential vs. Electric Field

Electric potential and electric field are related but distinct quantities.

The electric field is the negative gradient of the electric potential:

$\vec{E} = -\nabla V$

What this means in practice: the electric field points in the direction of decreasing potential. If you place a positive charge in a region, it will naturally move from high potential to low potential, just like a ball rolls downhill.

Calculation of Electric Potential

To find the electric potential at a point near a point charge, follow these steps:

1. Identify the charge $q$ (in coulombs) and the distance $r$ (in meters) from the charge to the point of interest.
2. Plug both values into $V = \frac{kq}{r}$.
3. Solve for $V$. Pay attention to the sign of $q$ since it determines whether the potential is positive or negative.

Example: Find the electric potential 2 meters away from a point charge of $3 \, \mu C$.

• $q = 3 \times 10^{-6}$ C
• $r = 2$ m
• $V = \frac{(8.99 \times 10^9)(3 \times 10^{-6})}{2} = 13{,}485 \text{ V}$

That's about 13.5 kV, which is a large potential. This makes sense because even microcoulombs of charge produce significant voltages at short distances.

Work and Potential Difference

The potential difference ($\Delta V$) between two points is the change in electric potential from one point to another. The work done by the electric field to move a charge $q_0$ between two points is:

$W = q_0 \Delta V = q_0 (V_B - V_A)$

This connects potential (which describes the field) to energy (which describes what happens to charges moving through that field).

Because electric potential is a scalar, the superposition principle applies directly: to find the total potential at a point due to multiple charges, just add up the individual potentials. No vector components needed. For example, if two point charges create potentials of $+500$ V and $-200$ V at the same location, the total potential there is simply $+300$ V.