Question

Two identical objects each hold a net charge of q = 2e. If the gravitational force between the two objects exactly cancels the electrostatic force between the objects, what is the mass of each object?

Answer 1

The two objects both have a net charge of 2e, where e is the charge of the electron. The electroncharge is the smallest possible charge (we say that charge is quantized ).

The two objects are identical, so they both have a mass m, and the same net charge q.
Now, the gravitational force between them is attractive. The force is given by

F = Gmm/r^2.

Here, G is the universal gravitational constant (6.67*10^-11), m is the mass of the objects, and r is the distance between them. Normally, the law has the form G mM/r^2, in the case of two different objects with masses M and m.

THe electrostatic force is repulsive (the two charges are the same, both negative because the electroncharge is negative... the charges have the same sign), and is given by

F = k*qq/r^2.

Here, k is a constant (1/(4*Pi*epsilon_0)), q is the charge of both objects, and r is again the distance in between them.


Now it's given that these forces are equal, i.e:

Gm^2/r^2 = k q^2/r^2

Using this, we can find the mass m of both objects. The r's cancel out:

Gm^2 = kq^2.

So the mass is given by

m^2 = kq^2/G ,

so m = Sqrt[kq^2/G] = q * Sqrt[k/G] = 2e*Sqrt[k/G] .

Now you just have to fill in the numbers:
e = 6.6*10^-19
k = something, don't know the number
G = 6.67*10^-11