# Electric force vs. gravity

A learner asked,

“In the video on Coulomb’s Law, why does Sal say the gravitational force is a weak force compared to the electrostatic force, even though it’s operating at such a large scale?”

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

Explained by Willy McAllister.

The reason you don’t feel electric force in everyday life is because almost every electron in the universe is nestled up close to an identical 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. Let’s compare the force of gravity to the electric force between two apples.

## Gravitational force between two apples

Suppose you have 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.

You already know the gravitational attraction between two apples. It’s practically nothing. You can compute the tiny force using the gravitational equation with $m_{1,2} = 100\,\text{grams}$ and $r = 1\,\text{meter}$.

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

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

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

That’s 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 a $+1\,\text C$ apple. Then put that electron over on the other apple to make a $-1\,\text C$ apple. (See the Background section below.)

You get a gigantic force of attraction. Electric force is computed with Coulomb’s Law. Compute the force between two $1$ coulomb charges placed $1$ meter apart,

$|\vec F| = K \dfrac{q_0 \,q_1}{r^2}\qquad$ where $K = 9 \times 10^9\, \text{newton-meter}^2/\text{coulomb}^2$

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

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

This is the force you would feel if a fully loaded oil supertanker was sitting on your head.

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

## Background

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

$1\,\text{mole}$ 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 water molecules.

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

If you take $1$ 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 $0.25\,\text{pounds}$.

$2000 \,\text{pounds} = 1\,\text{ton}$

$9\times 10^9\,\text{newtons}$ is about $1{,}100{,}000\,\text{tons}$, the weight of a fully loaded LR2 supertanker.