Electric fields (3.7.3)
On this page
Coulomb’s law states that any two point charges exert electrostatic forces on one another that are directly proportional to the product of their charges, and inversely proportional to the square of the distance between them.
Where:
- is the electrostatic force,
- and is the charge of each respective point charge, and
- is the separation distance.
Coulomb’s law applies to point charges, but can be valid for extended objects such as spheres. Spheres must be spherically symmetric, and the distance between the centres of two spheres must be much greater than their radii: essentially modelling them as point charges.
Coulomb’s law also only applies for stationary charges. If the charges are moving, then this introduces additional magnetic forces.
The law also assumes there are no external electric fields or other forces influencing the charges. If external fields are present, the resultant force must account for these additional interactions.
The diagram below shows the direction of the electrostatic forces between two point charges that have opposite charge and like charge:
Both charges in each case experience the same force due to Newton’s third law, which states that the charges will exert equal and opposite forces on one another.
Coulomb’s law states that the force between two point charges with charges and , separated by a distance is given by the equation:
Where:
- is the electrostatic force,
- and is the charge of each respective point charge, and
- is the separation distance.
- The constant of proportionality is the Boltzmann constant.
In Coulomb’s law, the constant of proportionality that is the Boltzmann constant, may be written in terms of the permittivity of free space . This allows one to rewrite Coulomb’s law as:
The permittivity of free space is a fundamental physical constant. It characterises the ability of an electric field to form and propagate throughout a vacuum, and how the electric fields interact. A greater value for the permittivity of free space would mean a weaker electric field for the same charges and distances.
Question walkthrough
Separation of Two Repelling Point Charges
Calculates the separation between two repelling point charges given their charge magnitudes and the force between them, using Coulomb's law.
A point charge or charged metal sphere produces a radial electric field. Field lines become less dense with increasing distance from the source, meaning the strength of the field decreases with distance from the charge.
The electric field strength at a distance from the centre of the sphere is equal to the electrostatic force divided by the charge. One can substitute Coulomb’s law for the force to obtain:
The electric field strength is directly proportional to the charge and is inversely proportional to the square of the distance , i.e. the strength decreases with distance.
The electric field for a charged metal sphere and a point charge decreases with distance.
The electric field strength is inversely proportional to the square of the distance from the centre. Therefore, if we plot the electric field strength against the reciprocal of the square of the distance from the centre we obtain a linear relationship:
As can be seen, the gradient is constant as it is a straight line relationship, and is proportional to the charge Therefore, one could obtain the magnitude of the charge from a plot of against
Question walkthrough
Electric Field Strength from a Point Charge
Calculates the electric field strength at a given distance from a point charge using Coulomb's law.
Electric fields produced by point charges have infinite range. The strength of the electric field due to a point charge follows an inverse square law with the distance from the point charge :
We see that the electric field strength is inversely proportional to the square of the distance from the point charge.
Electric fields from point charges are radial in nature: the field strength decreases radially outwards from the point charge.
Electric field lines are used to specify the direction of the field. Point charges produce radial electric fields.
The field lines point outwards for positive charges and inwards on negative charges.
A point charge and a uniformly charged sphere both produce a radial field. The uniformly charged sphere can be modelled as a point charge at its centre.
The field lines for the uniformly charged sphere (right) are the same as those for the point charge (left) beyond the dashed sphere that represents the edge of the charged sphere.
Due to the point charge and the uniformly charged sphere producing a radial field, we see that the electric field strength decreases with distance from the point charge and the uniformly charged sphere. The image shows that the space between the field lines increases with increasing distance from the point charge and uniformly charged sphere, indicating the field strength is decreasing.
Since the space between the field lines – and therefore the field strength – is decreasing, the field of a point charge and a uniformly charged sphere is non-uniform.
The electric field strength of an electric field at a point in space is defined as the force per unit charge experienced by a positive test charge at that point. It is a measure of the intensity of the electric field at that point and informs us how much force a positive test charge would experience within that field at that point.
A positive test charge is a hypothetical charge assumed to be positive, used to measure the strength and direction of an electric field at a particular point. It is positive by convention, so that the direction of the electric field lines aligns with the direction a positive charge would move.
The equation for the electric field strength is given by:
Where:
- is the force experienced by the positive test charge,
The unit of electric field strength is the newton per coulomb
The electric field strength is a vector quantity, possessing both magnitude and direction.
By convention, the direction of an electric field at a point in space is the direction a positive charge would move due to a force if placed at that point, as shown in the figure below.
Question walkthrough
Field Strength and Direction from Force on an Electron
Calculates the electric field strength from the force experienced by a moving electron, and determines the direction of the field.
When two parallel conducting plates have a potential difference applied across them, a uniform electric field is created in the space between them, pointing from the positive plate to the negative plate.
It is important to note that the electric field strength between two oppositely charged parallel plates is related to the potential difference, applied across them and their separation,
A uniform electric field is represented by parallel, evenly spaced electric field lines. The image below shows a positive test charge placed between two oppositely charged plates:
The positive point charge (blue circle) in the image experiences a constant force and gains kinetic energy as it travels from the positive plate to the negative plate along the electric field lines.
The equation for the electric field strength between two oppositely charged parallel plates can be expressed in terms of the potential difference, between the two plates.
A charge experiences a force when moving between two oppositely charged parallel plates. Work is done on the charge and is equal to the force and the distance the charge moves which is the plate separation.
It is important to note that since the definition of potential difference is the work done per unit charge, we can write the electric strength between two oppositely charged parallel plates as:
The units for the electric field strength are The above equation shows the units for electric field strength are also
This equation is only applicable to oppositely charged parallel plates.
When drawing the electric field lines between two oppositely charged conducting plates, ensure that the field lines:
Electric fields can have different configurations depending on the size, shape, charge, and arrangement of the objects that produce them.
Uniform field
- A uniform field has equally spaced parallel field lines
- Can be produced between two parallel charged plates
The table below highlights the differences in non-uniform electric fields:
| Feature | Opposite charges | Like charges |
|---|---|---|
| Charge configuration | Two point charges of equal magnitude | Two point charges of equal magnitude, same sign |
| Field-line pattern | Lines emerge from the positive charge and terminate on the negative charge, forming continuous curves between them | Lines emerge from (or terminate on) each charge and curve away from the other; no lines pass directly between the two charges |
| Field at the midpoint (superposition) | Contributions from each charge point in the same direction and add, giving an enhanced field directed from to | Contributions from each charge are equal in magnitude but opposite in direction; they cancel exactly, so the net field is zero |
| Uniformity | Non-uniform: magnitude and direction vary with position; strength falls with distance from each charge | Non-uniform: magnitude and direction vary with position; strength falls with distance from each charge |
| Point of zero net field between the charges | None. The two contributions reinforce everywhere along the axis between the charges | Present at the midpoint (for equal magnitudes). A neutral point |
Question walkthrough
Explaining Why Terminal Voltage Is Less Than EMF
Explain why the terminal voltage of a cell in series with a resistor is less than its EMF, in terms of internal resistance and energy loss.
Question walkthrough
Calculating Lost Voltage Due to Internal Resistance
Calculate the lost voltage across a battery's internal resistance, given the EMF, external resistance, and current from the ammeter reading.
Electric fields can be used to accelerate charged particles. For example, linear accelerators utilise electric fields to accelerate protons to speeds approaching the speed of light.
The image below shows two oppositely charged plates with a proton placed between them:
The proton, being positively charged, moves downwards following the electric field lines. As the force is constant, the proton has constant acceleration.
When a charged particle travels into a uniform electric field at a right angle, the particle follows a parabolic trajectory.
The image below shows the motion of a positively charged particle entering a uniform electric field at a right angle:
It is important to note that the particle follows a curved trajectory whilst inside the field and continues to follow a straight path once it has left the field.
There is no horizontal acceleration since no horizontal force is being applied, thus the horizontal component of the velocity remains constant throughout the motion.
There is a vertical acceleration of the particle due to the electric field applying a vertically downward force on the particle. The particle initially has zero vertical velocity but upon exiting the field has acquired a vertical component of velocity .
The increase in the vertical velocity of a particle moving through a uniform, vertical electric field with horizontal velocity, can be calculated using Newton’s Second Law of Motion.
The time spent within the field is equal to the length of the electric field region, divided by the particle’s horizontal velocity .
Therefore, the equation for the vertical velocity component of a charged particle moving through a uniform, vertical electric field :
Where:
- is the electric field strength,
- is the partcle’s charge,
- is the mass of the particle, and
- is the length of the electric field region.
If the particle has an initial vertical velocity component equal to then the final vertical velocity is:
Question walkthrough
Deriving Vertical Velocity in an Electric Field
Derive an expression for the vertical velocity gained by a charged particle in a uniform electric field, combining Newton's second law, the electric field strength equation, and the SUVAT equations of motion.
Question walkthrough
Proton Deflection in a Uniform Electric Field
Calculate the vertical velocity component gained by a proton deflected in a uniform electric field between charged parallel plates, using the electric field strength, plate separation, and initial horizontal velocity.
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.