Electric force vs. gravity

The gravitational force is extremely weak compared to the electric force.

Why does Sal say (at 4:40) the gravitational force is weak compared to the electric force?

Written by Willy McAllister.

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You don’t feel electric force in everyday life because almost every negative charge (electron) in the universe is nestled up close to a positive charge (the nucleus of an atom). That equalizes (neutralizes) the electric force. That’s why we are not aware of it most of the time.

Thought experiment: compare the force of gravity to the electric force between two apples.

Gravitational force between two apples

A medium sized apple has a volume of about $100$ cubic centimeters and weighs roughly $100\,\text{grams}$ (about $1/4$ pound). If you hold an apple in your hand the downward force you feel is about $1$ newton. That’s the force of attraction between the apple and the Earth. It is not too hard to lift an apple off the table. You are easily strong enough to overcome the gravitational attraction between the apple and our planet.

What about the gravitational attraction between an apple and another apple? It’s practically nothing. You can compute the tiny force using the Law of Gravity,

$F = G \,\dfrac{m_1 \, m_2}{r^2}\qquad$ where $G = 6.67 \times 10^{-11}\,\text N \, \text m^2/\text{kg}^2$

Set $m_1$ and $m_2$ to $100\,\text{grams}$. Place the apples $1\,\text{meter}$ apart, $r = 1\,\text{meter}$.

$F = 6.67 \times 10^{-11} \times \dfrac{0.1 \times 0.1}{1^2}$

$F = 6.67 \times 10^{-13}\,\text N$

The attraction between two apples is really close to $0$ force.

Electric force between two apples

Now for the electric force. The electric force between apples is $0$. That’s because there are equal numbers of $+$ and $-$ charges in both apples and everything is electrically neutral.

That’s not really a fair comparison with gravity, because the plus and minus charges are all intermingled. Let’s charge one apple up to $+1$ coulomb and the other to $-1$ coulomb. $1$ coulomb is the amount of charge that moves past a point in a wire in $1\,\text{second}$ when the current is $1\,\text{ampere}$. That’s a reasonable everyday current.

To charge an apple up to a coulomb you could (conceptually) remove $1$ electron for every $55{,}000$ water molecules, leaving an apple with a $+1\,\text C$ charge. Take that electron and put it on the other apple to give it a $-1\,\text C$ charge. (See the Background section below.)

What is the force of attraction? It is gigantic! We compute the electric force with Coulomb’s Law. The force between two $1$ coulomb charges placed $1$ meter apart is,

$|\vec F| = K \dfrac{q_0 \,q_1}{r^2}$

$K = 9 \times 10^9\, \text{newton-meter}^2/\text{coulomb}^2$

$|\vec F| = 9 \times 10^9 \times \dfrac{1 \times 1}{1^2}$

$|\vec F| = 9 \times 10^9 \,\text{newtons}$ (attracting)

This is the force you would feel if ten fully loaded oil supertankers were sitting on your head.

The electric force is unimaginably greater than the force of gravity.

Background

The mole is defined by Avogadro’s Number, $6.022 \times 10^{23}$ particles.

$1\,\text{mole}$ of $\text H_2\text O$ weighs $18\,\text{grams}$. $2\,\text{grams}$ of hydrogen plus $16\,\text{grams}$ of oxygen.

An apple is mostly water. $100\,\text{grams}$ of water is $100\,\text g/(18\,\text g/\text{mole}) = 5.5\,\text{moles}$ of mostly water molecules.

There are almost exactly $10{,}000\,\text{coulombs}$ of negative charge in $1\,\text{mole}$ of electrons.

If you remove $1$ electron from $1$ out of every $55{,}000$ molecules in an apple, it adds up to $1\,\text{coulomb}$ of electrons.

$1\,\text{newton}$ is about $0.1\,\text{kilogram}$
$1000 \,\text{kilograms} = 1\,\text{metric tonne}$
$1\,\text{tonne} \approx 10^4\,\text{newtons}$
$9\times 10^9\,\text{newtons}$ is about $9\times 10^9/10^4 = 900{,}000\,\text{tonnes}$.
A fully loaded LR2 supertanker weighs about $90{,}000\,\text{tonnes}$.