Similar but different (T)

Estimate the ratio of the gravitational force between two electrons and the electrostatic (Coulomb) force between two electrons.

The force which the sun exerts on the earth is entirely dominated by their gravitational interaction. What can you deduce from this?

[The electron charge e and mass m have ratio : e/m=1.76×1011Ckg-1.]

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Hint

It looks like the question should have specified the separation of the electrons. But it doesn’t need to.
 

Solution

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Solution

The gravitational force between two equal masses, m1=m2=m, separated by distance r is of magnitude (Key Point 2.4)

\[ F_G =G \frac{m^2}{r^2} \]
where G is the universal gravitational constant with value
G=6.67×10-11Nm2kg-2

The electrostatic (Coulomb) force between two equal charges, q1=q2=e, separated by distance r is of magnitude (Key Point 2.4)

\[ F_E = K\frac{e^2}{r^2} \]
where K (it might be called Coulomb’s constant, but it isn’t) has the value:
K=9.0×109Nm2C-2

Both forces are of inverse-square form: their strength varies inversely as the square of the separation.

Taking the ratio of the two forces the separation r cancels to leave:

\[ \frac{F_G}{F_E} = \frac{G}{K} \times \left( \frac{e}{m}\right)^{-2} = \frac{6.67 \times 10^{-11}}{9.0 \times 10 ^9 \times (1.76 \times 10^{11})^2} =2.4 \times 10^{-43} \hspace{0.5cm}\mbox{\rm (2sf)}\hspace{0.5cm} \]

The gravitational force acting between any pair of electrons is tiny by comparison with the electrostatic force. The same is true of any pair of protons. These statements are independent of separation. Nevertheless it is the gravitational force which dominates the interactions between large lumps of matter —in particular, bits of the solar system. The reason is that lumps of matter contain virtually identical numbers of electrons and protons. The total electric force between the like particles (electro-electron and proton-proton) is repulsive and cancels the total attractive force between unlike particles.

You might try estimating how closely the number of electrons and the number of protons in your body must match up if you are to be able to hug your mother, and choose when to do so.