Comparing gravitational fields and electrical fields Revision Notes for AQA A-Level Physics

Comparing gravitational fields and electrical fields

Gravitational and electric fields share many mathematical similarities despite describing fundamentally different physical phenomena. Understanding these parallels and differences provides insight into field theory and allows you to apply similar analytical approaches to both types of fields.

Fundamental equations

The mathematical descriptions of gravitational and electric fields follow analogous patterns. Both fields can be characterized by their forces, field strengths, potentials, and the work done in moving objects through them.

Force laws

Gravitational force between two masses is given by:

$F = \frac{Gm_1m_2}{r^2}$

where $G$ is the gravitational constant, $m_1$ and $m_2$ are the masses, and $r$ is their separation.

Electric force between two charges is given by:

$F = \frac{Q_1Q_2}{4\pi\varepsilon_0 r^2}$

where $Q_1$ and $Q_2$ are the charges, $\varepsilon_0$ is the permittivity of free space, and $r$ is their separation.

Both forces follow an inverse square law with distance. The gravitational force is always attractive, whereas the electric force can be attractive (opposite charges) or repulsive (like charges).

infoNote

The key difference in force behaviour stems from the nature of the source properties: mass is always positive, but charge can be positive or negative. This fundamental distinction explains why gravitational forces only attract, while electric forces can either attract or repel.

Field strength

Gravitational field strength is defined as the force per unit mass:

$g = \frac{F}{m}$

For a radial field around a point mass $M$ :

$g = \frac{GM}{r^2}$

Electric field strength is defined as the force per unit charge:

$E = \frac{F}{Q}$

For a radial field around a point charge $Q$ :

$E = \frac{Q}{4\pi\varepsilon_0 r^2}$

Both field strengths decrease with the square of distance from the source, following the inverse square law.

lightbulbExample

Worked Example: Comparing Field Strengths

Consider a point mass of $6.0 \times 10^{24}$ kg and a point charge of $+2.0 \times 10^{-6}$ C, both measured at a distance of 1.0 m.

Gravitational field strength: $g = \frac{GM}{r^2} = \frac{(6.67 \times 10^{-11})(6.0 \times 10^{24})}{(1.0)^2} = 4.0 \times 10^{14} \text{ N kg}^{-1}$

Electric field strength: $E = \frac{Q}{4\pi\varepsilon_0 r^2} = \frac{2.0 \times 10^{-6}}{4\pi(8.85 \times 10^{-12})(1.0)^2} = 1.8 \times 10^{4} \text{ N C}^{-1}$

Both follow the $r^{-2}$ relationship, so doubling the distance would reduce each field strength to one quarter of its value.

Potential

Gravitational potential at a point represents the work done per unit mass in bringing a mass from infinity to that point:

$V = -\frac{GM}{r}$

The negative sign indicates that work is done by the field (gravitational force is attractive).

Electric potential at a point represents the work done per unit charge in bringing a small positive test charge from infinity to that point:

$V = \frac{Q}{4\pi\varepsilon_0 r}$

Electric potential is measured in volts and is a scalar quantity. The sign depends on whether the source charge is positive or negative.

infoNote

Both potentials are inversely proportional to distance from the source ( $1/r$ relationship). This is different from field strength, which follows a $1/r^2$ relationship. The shallower gradient means potential changes more gradually with distance than field strength does.

chatImportant

Sign Conventions for Potential

Gravitational potential is always negative (approaching zero at infinity) because gravitational forces are always attractive. The negative sign in $V = -GM/r$ ensures this.

Electric potential can be positive or negative depending on the source charge. A positive charge creates positive potential, while a negative charge creates negative potential. The sign matters when calculating work done or potential energy.

Work done

The work done in moving an object through a potential difference $\Delta V$ is:

Gravitational: $\Delta W = m\Delta V$

Electric: $\Delta W = Q\Delta V$

The work done depends on the potential difference and the mass or charge being moved.

Potential gradient

The relationship between field strength and potential gradient is:

Gravitational:

$g = -\frac{\Delta V}{\Delta r}$

Electric:

$E = -\frac{\Delta V}{\Delta r}$

The magnitude of the field strength equals the gradient of the potential-distance graph. The negative sign in the gravitational case reflects the convention that gravitational potential increases (becomes less negative) with distance.

infoNote

The negative sign appears in both equations because field strength points in the direction of decreasing potential. When moving in the direction of the field, potential decreases, giving a negative gradient. The field strength itself is defined as a positive quantity pointing in this direction.

Similarities between gravitational and electric fields

Both gravitational and electric fields exhibit several common characteristics that reveal their underlying mathematical structure:

Inverse square force laws: The gravitational force between masses and the electrostatic force between charges both vary inversely with the square of separation. This $r^{-2}$ dependence is a fundamental property of both fields.

Proportionality to source properties: Gravitational force is directly proportional to the product of the two masses ( $m_1m_2$ ). Electric force is directly proportional to the product of the two charges ( $Q_1Q_2$ ). This symmetry in the force laws reflects the similar mathematical structure.

Field line representation: Both gravitational and electric fields can be represented using field lines. These lines indicate the direction of force on a test mass or test charge placed in the field. The density of field lines represents the strength of the field at that location.

Inverse square field strength: Both gravitational field strength near a point mass and electric field strength near a point charge vary inversely with the square of distance ( $1/r^2$ ). This means doubling the distance reduces the field strength to one quarter of its original value.

infoNote

The inverse square relationships for both force and field strength are not coincidental—they emerge from the geometry of three-dimensional space. Any influence that spreads uniformly in all directions from a point source must follow an inverse square law because the surface area of a sphere increases as $4\pi r^2$ .

Potential definitions: Gravitational potential at a point is the work done per unit mass in bringing a mass from infinity to that point. Electric potential at a point is the work done per unit charge in bringing a positive test charge from infinity to that point. Both definitions follow the same conceptual pattern.

Inverse relationship with distance: Both gravitational and electric potentials are inversely proportional to distance from the source ( $1/r$ ). This contrasts with the field strengths, which follow a $1/r^2$ relationship.

Equipotential surfaces: Both fields can be represented by equipotential surfaces, which are perpendicular to field lines. No work is done when moving a mass or charge along an equipotential surface, as there is no potential difference.

lightbulbExample

Worked Example: Equipotential Surfaces

Consider a point charge of $+5.0 \times 10^{-9}$ C. Calculate the electric potential at distances of 0.10 m, 0.20 m, and 0.40 m from the charge.

At r = 0.10 m: $V = \frac{Q}{4\pi\varepsilon_0 r} = \frac{5.0 \times 10^{-9}}{4\pi(8.85 \times 10^{-12})(0.10)} = 450 \text{ V}$

At r = 0.20 m: $V = \frac{5.0 \times 10^{-9}}{4\pi(8.85 \times 10^{-12})(0.20)} = 225 \text{ V}$

At r = 0.40 m: $V = \frac{5.0 \times 10^{-9}}{4\pi(8.85 \times 10^{-12})(0.40)} = 112.5 \text{ V}$

Each calculated distance represents an equipotential surface—a sphere centered on the charge. Notice how doubling the distance halves the potential, demonstrating the $1/r$ relationship.

Differences between gravitational and electric fields

Despite their similarities, gravitational and electric fields have important differences that stem from the fundamental nature of mass and charge:

Nature of source: Mass is always positive—there are no negative masses in nature. Electric charge, however, exists in both positive and negative forms. This fundamental difference affects the behaviour of the fields.

chatImportant

The absence of negative mass means we cannot create regions where gravitational fields cancel out or produce repulsion. In contrast, electric fields can be arranged to cancel completely, and we can create both attractive and repulsive configurations. This makes electric fields far more versatile for practical applications.

Direction of force: Gravitational force is always attractive between any two masses. Electric force can be either attractive (between opposite charges) or repulsive (between like charges), depending on the polarity of the charges involved.

Medium dependence: The gravitational force between masses is independent of the medium between them—it has the same value whether the masses are in a vacuum, air, or any other material. The electric force between charges is affected by the material between them. The permittivity $\varepsilon_0$ in the equation applies only to a vacuum; in other materials, a different permittivity value must be used.

infoNote

When charges are placed in a material (dielectric), the electric force between them is reduced by a factor called the relative permittivity (or dielectric constant) $\varepsilon_r$ . The force becomes:

$F = \frac{Q_1Q_2}{4\pi\varepsilon_0 \varepsilon_r r^2}$

For example, water has $\varepsilon_r \approx 80$ , so the electric force between charges in water is about 80 times weaker than in a vacuum.

bookmarkSummary

Key Points to Remember:

Both gravitational and electric fields follow inverse square laws for force ( $1/r^2$ ) and inverse laws for potential ( $1/r$ ).
Gravitational force is always attractive; electric force can be attractive or repulsive depending on charge polarity.
Field strength equals the gradient of the potential-distance graph: $g = -\Delta V/\Delta r$ for gravity, $E = -\Delta V/\Delta r$ for electric fields.
Both fields can be represented by field lines and equipotential surfaces, which are always perpendicular to each other.
Gravitational force is independent of the medium, while electric force depends on the permittivity of the material between charges.
The mathematical parallels between the two fields allow you to apply similar problem-solving techniques, but always remember the fundamental physical differences in their sources and behaviour.

Comparing gravitational fields and electrical fields (AQA A-Level Physics): Revision Notes