Gravitational potential and spherical shells

Potential: energy per kilogram

The gravitational potential at a point is the potential energy of one kilogram placed there:

in J/kg. It is a scalar, so the potentials of several masses simply add. The field is the downhill slope of the potential: $g = -\dfrac{dV}{dr}$.

A hollow shell

Newton proved two remarkable results for a thin, uniform spherical shell of mass $M$ and radius $a$:

• Outside, it pulls as if all its mass were at its centre: $V = -\dfrac{GM}{r}$.
• Inside, the field is zero everywhere, so the potential is the same everywhere: $V = -\dfrac{GM}{a}$.

Inside, the near part of the shell is closer but has less mass in any cone of directions; the far part is farther but has more. With an inverse square law, they cancel exactly.

You should find the potential flat inside the shell: no slope, so no field, and nothing would push you anywhere.

Watch out. Zero field inside does not mean zero potential. The potential inside is negative, the same as at the shell.

A worked example

What is the Earth's potential at the Moon's distance, $3.84 \times 10^{8}\ \text{m}$? ($GM = gR^2 = 4.01 \times 10^{14}\ \text{m}^3/\text{s}^2$.)

1. Formula. $V = -\dfrac{GM}{r}$.
2. Numbers. $V = -\dfrac{4.01 \times 10^{14}\ \text{m}^3/\text{s}^2}{3.84 \times 10^{8}\ \text{m}} = -1.04 \times 10^{6}\ \text{J/kg}$.
3. Compare. At the surface it is $-6.27 \times 10^{7}\ \text{J/kg}$: sixty times deeper.

Think first. Potentials add as numbers; fields add as arrows. Choose the one that makes the problem easier.

Practise with new numbers

Try it in Python

The program works out the potential and the field of a shell, inside and outside.

Remember these

• $V = -\dfrac{GM}{r}$, in J/kg; potentials add as numbers.
• Shell, outside: like a point mass at the centre.
• Shell, inside: zero field, constant potential $-\dfrac{GM}{a}$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Two equal masses are $2d$ apart. Is the field zero midway? Is the potential?

2. An astronaut floats inside a huge hollow spherical asteroid of uniform thickness. Which way does the asteroid pull her?

3. How does the potential change as you move from $2R$ to $4R$ from a planet's centre?

4. What is the unit of gravitational potential?