Gravitational potential and comparison with electric fieldsEdexcel A-Level Physics: Revision notes

The gravitational potential $V$ at a point is the work done per unit mass in moving a small mass from infinity to that point. It is measured in J kg⁻¹. It is a scalar, and it is defined to be zero at infinity.

Because gravity is attractive, the field does positive work as a mass moves in from infinity, so the potential at every finite distance is negative. Equivalently, work must be done on a mass to take it out to infinity.

For a point mass or uniform sphere of mass $M$, at a distance $r$ from its centre:

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

The closer to the mass, the more negative $V$ becomes, so the potential increases as you move away from the mass.

A mass $m$ at a point of potential $V$ has gravitational potential energy $E = mV$. When a mass moves between two points, the change in potential energy is:

$\Delta E = m\,\Delta V$

If $\Delta V$ is positive (moving away from the mass) the object gains energy; the work done by an external force equals $m\Delta V$.

Potentials from several masses add as scalars, which is easier than adding fields as vectors.

Worked example: Raise 450 kg from the Earth's surface ($r = R$) to $r = 2R$. $\Delta V = \frac{GM}{R} - \frac{GM}{2R} = \frac{GM}{2R} = 3.13\times10^{7}$ J kg⁻¹, so $\Delta E = 450\times3.13\times10^{7} = 1.41\times10^{10}$ J.

Both fields describe forces acting at a distance, and they share a mathematical form:

• Both are radial around a point source, with field lines pointing towards or away from it
• Both forces obey an inverse square law: $F = Gm_1m_2/r^2$ and $F = Q_1Q_2/4\pi\varepsilon_0 r^2$
• Field strength is force per unit of the property that feels the field: $g = F/m$ in N kg⁻¹, $E = F/Q$ in N C⁻¹
• Field strength falls as $1/r^2$: $g = GM/r^2$ and $E = Q/4\pi\varepsilon_0 r^2$
• Potential is zero at infinity, with $V = -GM/r$ and $V = Q/4\pi\varepsilon_0 r$

The analogy is: mass $\leftrightarrow$ charge and $G \leftrightarrow 1/4\pi\varepsilon_0$.

• Gravitational force is always attractive; electric force can be attractive or repulsive, because there are two types of charge
• Gravitational potential is always negative; electric potential can be positive (near a positive charge) or negative (near a negative charge)
• Gravity depends on mass; the electric force depends on charge, and a body can be neutral
• Gravity is far weaker: the electric force between a proton and an electron is about $10^{39}$ times larger than the gravitational force
• Electric fields can be shielded by conductors or screened by opposite charges; gravitational fields cannot

Because planets and stars are almost electrically neutral, electric forces cancel and gravity dominates at large scales.

A proton and an electron are $5.3\times10^{-11}$ m apart.

Gravitational: $F = \frac{6.67\times10^{-11}\times9.11\times10^{-31}\times1.67\times10^{-27}}{(5.3\times10^{-11})^2} = 3.6\times10^{-47}$ N.

Electric: $F = \frac{(1.60\times10^{-19})^2}{4\pi\times8.85\times10^{-12}\times(5.3\times10^{-11})^2} = 8.2\times10^{-8}$ N.

The ratio of electric to gravitational force is about $2\times10^{39}$, and since both forces are inverse square the ratio does not depend on separation. This is why gravity is ignored in atomic physics.