Electric Field and Gravitational Field Comparison

Understanding the similarities and differences between electric and gravitational fields is more than an academic exercise—it reveals a profound unification in how we model fundamental forces in nature. For A-Level Physics, this comparison is crucial: it deepens your grasp of field theory, streamlines problem-solving by highlighting parallel mathematical structures, and is a common source of examination questions that test your ability to think analogically.

Force: The Fundamental Starting Point

Both electric and gravitational forces are described by remarkably similar equations, establishing them as action-at-a-distance forces mediated by their respective fields. The gravitational force between two point masses is governed by Newton's Law of Universal Gravitation:

$$F_g=G\frac{m_1{m}_2}{r^2}$$

Here, $F_g$ is the gravitational force, $G$ is the gravitational constant, $m_1$ and $m_2$ are the masses, and $r$ is the separation between their centres.

The electric force between two point charges is given by Coulomb's Law:

$$F_e=\frac{1}{4\pi {ϵ}_0}\frac{q_1{q}_2}{r^2}$$

Here, $F_e$ is the electrostatic force, $\frac{1}{4\pi {ϵ}_0}$ is the Coulomb constant, $q_1$ and $q_2$ are the charges, and $r$ is their separation.

The core similarity is the inverse square law dependence on distance ($1/r^2$). This means doubling the separation reduces the force to a quarter of its original strength for both gravity and electrostatics. The critical difference lies in the nature of the interaction: gravity is only ever attractive, as mass ($m$) is always positive. Electrostatic forces, however, can be either attractive or repulsive because charge ($q$) can be positive or negative. This leads to the rule: like charges repel, unlike charges attract.

Field Strength and Potential: Defining the Field Itself

We define a field to understand the influence an object exerts on the space around it. Gravitational field strength ($g$) at a point is defined as the force per unit mass experienced by a small test mass placed at that point: $g=F/m_{test}$. For a point mass $M$, it is derived from Newton's Law:

$$g=G\frac{M}{r^2}$$ The units are newtons per kilogram (N kg${}^{-1}$), and the direction is towards the mass $M$.

Similarly, electric field strength ($E$) is defined as the force per unit positive charge: $E=F/q_{test}$. For a point charge $Q$, it is:

$$E=\frac{1}{4\pi {ϵ}_0}\frac{Q}{r^2}$$ The units are newtons per coulomb (N C${}^{-1}$). The direction is radially away from a positive $Q$ and towards a negative $Q$.

Again, both follow the inverse square law. The field strength tells you about the force a test object would feel. A more advanced concept is potential. Gravitational potential ($V_g$) at a point is the work done per unit mass to bring a small test mass from infinity to that point. For a point mass $M$:

$$V_g=-G\frac{M}{r}$$ Electric potential ($V_e$) is the work done per unit positive charge: $V_e=\frac{1}{4\pi {ϵ}_0}\frac{Q}{r}$.

Notice the key sign difference: gravitational potential is always negative (as you must do work against the field to move to infinity), while electric potential can be positive or negative depending on the sign of $Q$. Both potentials are scalar quantities, measured in J kg${}^{-1}$ and J C${}^{-1}$ (or volts) respectively.

Potential Energy and Work Done

The potential energy of a system is directly linked to potential. The gravitational potential energy ($E_p$) for two masses is:

$$E_p=-G\frac{m_1{m}_2}{r}$$ The electrostatic potential energy for two charges is:

$$E_p=\frac{1}{4\pi {ϵ}_0}\frac{q_1{q}_2}{r}$$

The work done to move an object within a field is equal to its change in potential energy. The sign conventions are vital. For gravity, moving a mass away from a planet increases $E_p$ (it becomes less negative). For electricity, moving a positive charge away from another positive charge also increases $E_p$ (it becomes less positive). The force does negative work when an object moves in the direction it is naturally attracted (e.g., a falling mass, or opposite charges coming together).

Worked Example Parallel: Calculate the force and potential energy for (a) two 5 kg masses 0.2 m apart, and (b) two +5 µC charges 0.2 m apart.

• (a) Gravitational: $F_g=(6.67\times 10^{-11})\frac{(5)(5)}{(0.2{)}^2}\approx 4.17\times 10^{-8}$ N. $E_p=-(6.67\times 10^{-11})\frac{(5)(5)}{0.2}\approx -8.34\times 10^{-9}$ J.
• (b) Electric: $F_e=(8.99\times 10^9)\frac{(5\times 10^{-6})(5\times 10^{-6})}{(0.2{)}^2}\approx 5.62$ N. $E_p=(8.99\times 10^9)\frac{(5\times 10^{-6})(5\times 10^{-6})}{0.2}\approx 1.12$ J.

This starkly shows the relative weakness of gravity and the effect of sign.

Interactions and Visualisations: Field Lines and Equipotentials

Both fields can be visualised using field lines (lines of force) and equipotential surfaces. Field lines show the direction of force on a test object. For a point mass or a positive point charge, they are radial lines pointing inwards (gravity) or outwards (positive charge). The density of lines indicates field strength. Equipotentials are surfaces where the potential is constant. They are always perpendicular to field lines. For a point source, they are spherical shells.

In a uniform field (like near the Earth's surface or between parallel charged plates), the field lines are parallel and equally spaced, and the equipotentials are parallel planes perpendicular to them. The closer the equipotentials, the stronger the field. A key skill is interpreting diagrams of combined fields, such as those around two point charges. You must recognise that the field is the vector sum of individual fields, and equipotentials bend around the charges, never crossing each other.

Common Pitfalls

1. Misapplying Signs in Electric Calculations: The most frequent error is forgetting that the sign of charge is crucial in Coulomb's Law, electric field, and potential energy. Always substitute charge with its sign into $F_e$ and $E_p$ equations. For force, the sign of the result indicates direction (negative for attraction). For $E_p$, a negative result means a bound, attractive system.

• Correction: When calculating magnitude of force, use absolute values of charge. When determining direction or the nature of $E_p$, always use signed charges.

1. Confusing Potential and Potential Energy: Students often treat $V$ and $E_p$ as interchangeable. Remember, potential ($V$) is a property of a point in a field (energy per unit mass/charge). Potential energy ($E_p$) is a property of a system of objects (e.g., two masses or two charges).

• Correction: Use $E_p=m{V}_g$ or $E_p=q{V}_e$ to connect them. The potential energy of a test object at a point is its charge/mass multiplied by the potential at that point.

1. Misinterpreting Field Line Diagrams for Electric Fields: Assuming field lines show the path a charged particle will take is incorrect. They show the direction of force, which is the same as the direction of acceleration, not necessarily velocity.

• Correction: A particle will accelerate tangentially to a field line, but if it has an initial velocity, it will not follow the line. Field lines indicate the force on a positive test charge.

1. Assuming Constants are of Similar Magnitude: Treating $G$ and $\frac{1}{4\pi {ϵ}_0}$ as numerically similar leads to wildly incorrect force comparisons, as shown in the worked example. Gravity is exceptionally weaker.

• Correction: Remember the orders of magnitude: $G\approx 10^{-11}$, $k\approx 10^9$. This difference is fundamental.

Summary

• Both gravitational and electric fields are modelled by inverse square law relationships for force, field strength, and potential ($\propto 1/r^2$ and $\propto 1/r$ respectively), leading to parallel mathematical structures.
• The fundamental difference is that gravitational interactions are attractive-only due to mass always being positive, while electric interactions can be attractive or repulsive due to charge having two signs.
• Field strength ($g$ and $E$) is a vector defining force per unit test quantity, while potential ($V_g$ and $V_e$) is a scalar defining work done per unit test quantity. Gravitational potential is always negative; electric potential's sign depends on the source charge.
• Problem-solving requires careful attention to sign conventions for electric fields and an awareness of the vast difference in the magnitudes of the fundamental constants $G$ and $\frac{1}{4\pi {ϵ}_0}$.
• Both fields are visualised with perpendicular sets of field lines (showing direction of force) and equipotential surfaces (showing contours of constant potential).