Question

Two electrically-charges spheres are suspended from insulated threads a certain distance from each other. There is a certain amount of electrostatic force between them. Describe specifically (not just increase or decrease) what happens to this force in each of the scenarios below
1.)The charge on one sphere is reduced by half
2.)The charge on both spheres is doubled
3.)The distance between the spheres is increased by a factor of three
4.)The distance between the sphere is decreased to one-fourth

Answer 1

You can figure out all of this just by looking at the equation for electrostatic force (Coulomb's Law):

\[F_0=\frac{kq_1q_2}{r^2}\]
I called the force F(0) just to indicate that it's our starting force. Now let's look at situation #1 - we half the magnitude of one charge:

\[F_1=\frac{k \left( \frac{1}{2}q_1 \right) q_2}{r^2}\]

Now just compare the two forces. The hard(ish) but official way is to set up a ratio:

\[\frac{F_1}{F_0}= \frac{\frac{k \left( \frac{1}{2}q_1 \right) q_2}{r^2}}{\frac{kq_1q_2}{r^2}}\]
Simplify this monstrosity and you get:
\[F_1=\frac{1}{2}F_0 \]
So the electrostatic force after you half the magnitude of one charge is halved as well. The easier way is to just look at the algebra itself without simplifying - the only difference between the equations is a factor of 1/2, so this translates directly to how the force changes.

Give the next few situations a shot using this method!