# Electric Potential Due to a Point Charge

## The electric potential of a point charge Q is given by $V=\frac{kQ}{r}$.

#### Key Points

• Recall that the electric potential is defined as the potential energy per unit charge, i.e. <equation contenteditable="false">$V=\frac{PE}{q}$.

• The potential difference between two points ΔV is often called the voltage and is given by $\Delta V = V_{B} - V_{A} = \frac{\Delta PE}{q}$ . The potential at an infinite distance is often taken to be zero.

• The case of the electric potential generated by a point charge is important because it is a case that is often encountered. A spherical sphere of charge creates an external field just like a point charge, for example.

• The equation for the electric potential due to a point charge is $V=\frac{kQ}{r}$  , where k is a constant equal to 9.0×109 N⋅m2/C2.

#### Terms

• The potential energy per unit charge at a point in a static electric field; voltage.

• The amount of electrostatic potential between two points in space.

#### Figures

1. ##### Van de Graaff Generator

The voltage of this demonstration Van de Graaff generator is measured between the charged sphere and ground. Earth’s potential is taken to be zero as a reference. The potential of the charged conducting sphere is the same as that of an equal point charge at its center.

## Electric Potential Due to a Point Charge

### Overview

Recall that the electric potential is defined as the electric potential energy per unit charge

$V=\frac{PE}{q}$

The electric potential tells you how much potential energy a single point charge at a given location will have. The electric potential at a point is equal to the electric potential energy (measured in joules) of any charged particle at that location divided by the charge (measured in coulombs) of the particle. Since the charge of the test particle has been divided out, the electric potential is a "property" related only to the electric field itself and not the test particle. Another way of saying this is that because PE is dependent on q, the q in the above equation will cancel out, so V is not dependent on q.

The potential difference between two points ΔV is often called the voltage and is given by

$\Delta V = V_{B} - V_{A} = \frac{\Delta PE}{q}$

### Point Charges

Point charges, such as electrons, are among the fundamental building blocks of matter. Furthermore, spherical charge distributions (like on a metal sphere, see figure below) create external electric fields exactly like a point charge. The electric potential due to a point charge is, thus, a case we need to consider. Using calculus to find the work needed to move a test charge from a large distance away to a distance of r from a point charge Q, and noting the connection between work and potential (W=–qΔV), it can be shown that the electric potential V of a point charge is

$V=\frac{kQ}{r}$ (point charge)

where k is a constant equal to 9.0×109 N⋅m2/C2 (Figure 1).

The potential at infinity is chosen to be zero. Thus V for a point charge decreases with distance, whereas E for a point charge decreases with distance squared:

$E=\frac{F}{q}=\frac{kQ}{r^{2}}$

The electric potential is a scalar while the electric field is a vector. Note the symmetry between electric potential and gravitational potential - both drop off as a function of distance to the first power, while both the electric and gravitational fields drop off as a function of distance to the second power.

#### Key Term Glossary

coulomb
In the International System of Units, the derived unit of electric charge; the amount of electric charge carried by a current of 1 ampere flowing for 1 second. Symbol: C
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electric field
A region of space around a charged particle, or between two voltages; it exerts a force on charged objects in its vicinity.
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electric potential
The potential energy per unit charge at a point in a static electric field; voltage.
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energy
A quantity that denotes the ability to do work and is measured in a unit dimensioned in mass × distance²/time² (ML²/T²) or the equivalent.
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equation
An assertion that two expressions are equal, expressed by writing the two expressions separated by an equal sign; from which one is to determine a particular quantity.
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matter
The basic structural component of the universe. Matter usually has mass and volume.
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particle
A very small piece of matter, a fragment; especially, the smallest possible part of something.
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potential
A curve describing the situation where the difference in the potential energies of an object in two different positions depends only on those positions.
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potential difference
The difference in potential energy between two points in an electric field; the difference in charge between two points in an electrical circuit; voltage.
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potential energy
The energy an object has because of its position (in a gravitational or electric field) or its condition (as a stretched or compressed spring, as a chemical reactant, or by having rest mass)
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power
A measure of the rate of doing work or transferring energy.
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scalar
A quantity that has magnitude but not direction; compare vector.
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Scalar
A quantity which can be described by a single number, as opposed to a vector which requires a direction and a number.
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symmetry
Exact correspondence on either side of a dividing line, plane, center or axis.
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vector
A directed quantity, one with both magnitude and direction; the between two points.
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voltage
The amount of electrostatic potential between two points in space.
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work
A measure of energy expended in moving an object; most commonly, force times displacement. No work is done if the object does not move.