physics help please?

Question

A charge of +1.80 x 10^-9 C is placed at the origin, and another charge of +5.00 x 10^-9 C is placed at x=1.9m. The coulomb constant is 8.98755 x 10^9 (N x m^2/c^2). Find the point (coordinate) between these two charges where a charge of +3.50 x 10^-9 C should be placed so that the net electric force on it is zero. Answer in units of meters.

Answer 1

Actually, you don't need the coulomb constant or even the size of the third charge to answer this question. (Or even the sizes of the original charges, as long as you know their ratio.) Neat, huh?

We want F1 = F2. Let the charge at the origin be q1, the charge at x = 1.9 m be q2, the remaining charge q3. Suppose this charge is at position x. Then the distance betqeen q1 and q3 is x m, and the distance between q2 and q3 is (1.9 - x) m.

So we get
k q1 q3 / x^2 = k q2 q3 / (1.9-x)^2
=> q1 / x^2 = q2 / (1.9-x)^2
=> (1.9-x) / x = √(q2/q1)
=> 1.9/x - 1 = 1.667
=> x = 1.9 / 2.667 = 0.71 m.

Answer 2

Force on unit charge at a point distant a from the 1.8 charge is

F1 = K 1.8 / a^2 where

Similarly the force on unit charge at a point disant b from 5 charge is

F2 = K 5 / b^2.

For equilibrium F1 = F2.

Therefore
1.8 / a^2 = 5/b^2

b/a = √( 5/1.8) = 1.67.

b= 1.67a

But a +b = 1.9m.

a + 1.67 a = 1.9m

a = 1.9 / 2.67 =.71 m

b =1.19m