Electric potential and energy (6.2.4)
On this page
It is important to note that the definition of electric potential at a point in space is the work done per unit charge to bring a positive test charge from infinity to that point.
If a positive test charge is far enough from a positive charge to feel practically no electric field, it can be said to be at infinity.
Work must be done to bring towards due to the electrostatic repulsion between them.
The electric potential of at point (at a distance from the charge ) equals the work done per unit charge to bring to point from infinity.
The electric potential is defined as zero at infinity. As a positive test charge approaches another positive charge, its electric potential increases due to electrostatic repulsion, as illustrated below.
Conversely, if a positive test charge is brought towards a negative charge, the electric potential decreases from zero to a negative value. Work must be done to move the positive test charge away from the negative charge due to the electrostatic attraction, as shown below.
The electric potential at a point is defined as the work done per unit charge in bringing a positive test charge from infinity to that point and is given by:
Where:
- is the charge,
- is the distance from the charge to the point at which the potential is measured,
- is a mathematical constant that comes from the way electric fields behave around a sphere, and
- is the permittivity of free space.
It is important to note that the equation does not depend on the test charge .
The units for electric potential are or volts .
It is important to note that the electric potential difference is defined as the work done per unit charge to move a positive test charge between two points in a particle’s electric field.
The electric potential difference between the two points A and B in the diagram above is the difference between the potentials at these points, i.e.
The work done to move a charge between two points in an electric field is equal to:
Where is the electric potential difference between the two points.
For a unit positive charge, , the work done is equal to the electric potential energy:
Question walkthrough
Finding Electric Potential Near a Nucleus
Calculate the charge of a uranium nucleus and use it to find the electric potential at a given distance from its surface.
A capacitor can store charge. An isolated charged sphere of radius is also able to store charge and, therefore, can be classed as a capacitor with a single plate.
A charged sphere’s capacitance is equal to the charge stored divided by the electric potential at the surface.
The equation for the capacitance of an isolated sphere is:
Where:
- is the permittivity of free space, and
- is a mathematical constant that comes from the way electric fields behave around a sphere.
The units for capacitance are Farads
It is important to note that outside an isolated charged sphere, the electric potential is equal to the electric potential of a point charge (with the same charge as the sphere) at the centre of the sphere.
Therefore, the electric potential at the surface of a charged sphere is:
Where:
- is the charge stored on the sphere,
- is the radius of the sphere,
- is the permittivity of free space, and
- is a mathematical constant that comes from the way electric fields behave around a sphere.
Question walkthrough
Finding Capacitance of an Isolated Sphere
Derive C = 4πε₀R for an isolated sphere and find its radius from a given volume to calculate capacitance.
A uniformly charged sphere may be treated as a point charge, with the force between two point charges given by Coulomb’s law:
Where:
- is the force,
- and are the charge of each point charge,
- is the permittivity of free space,
- is a constant related to the spherical behaviour of electric fields, and
- is the separation distance between the two point charges.
The electrostatic force between two point charges varies with the separation. The graph illustrates the relationship between the force and the separation:
It is important to note that, as the plot illustrates, the force and separation follow an inverse-square relationship, i.e., the force is proportional to the reciprocal of the separation squared.
Since the work done equals force multiplied by distance, this can be calculated by finding the area under the curve of a force–separation plot.
As shown, the area under the force–separation plot for two-point charges equals the work done. This is the work done to bring the two point charges from infinity to separation,
This total work done is equivalent to the electric potential energy given by:
The magnitude of the electric potential energy represents the amount of energy needed to completely separate the two point particles to infinity.
The force–separation relationship applies to both spherical charges and point charges.
Similarly to stretched springs possessing potential energy, charged particles can also possess potential energy. Consider the diagram below of two positive charges being brought closer together
The repulsive force increases as the separation distance of the two positive charges decreases.
It is important to note that work must be done to bring the charges closer together. As the charges move closer to each other, more work must be done per unit distance as the repulsive force increases.
The work done is stored as electric potential energy, which is released when the separation between the charges is increased.
The electric potential is defined as the work done per unit charge in bringing a positive test charge from infinity to that point.
Since the work done is equivalent to the electric potential energy then the electric potential energy is equal to the electric potential multiplied by the charge, :
The equation for electric potential is:
So, the electric potential energy required to bring a positive test charge to a point in the field of a charge is:
Question walkthrough
Finding Ionisation Energy of a Hydrogen Atom
Use the electric potential energy formula to find the ionisation energy of a hydrogen atom in electronvolts.