Two small identical conducting balls start with different amounts of charge on t

Question

Two small identical conducting balls start with different amounts of charge on them. Initially, when they are separated by 85.0 cm (this is the center-to-center distance), one ball exerts an attractive force of 2.10 N on the second ball. The balls are then touched together briefly (charge is conserved in this process), and then again separated by 85.0 cm. Now both balls have a positive charge, and the force that one ball exerts on the other is a repulsive force of 1.30 N. For this problem, use k = 9.0 109 N m2 / C2, and assume that the radius of each ball is small compared to 85.0 cm.

(a) After the two balls have been touched together, what is the charge on each ball?

(b) Before they were touched together, one ball had a positive charge. How much charge did that ball have?

(c) Before they were touched together, one ball had a negative charge. How much charge did that ball have? (Don't forget to include the minus sign!)

Explanation / Answer

Let the charge on the two balls be q and q' respectively. In this one charge is positive and another is negative. Since one ball exerts an attractive force of 2.10 N on the second ball. Now let q be positive and q' be negative. Attractive force F = 2.1 N, Separation distance r = 0.85 m.

As we know from Coulomb's law F = Kq(-q')/r2

So, -q q ' = F r 2 / K

Where K = Coulomb's constant = 9 x109 Nm2/C2

Now ,substituting values we get,

- q q ' = 1.686 x10 -10            -----------( 1)

After touching the balls ,the charge on each ball Q = (q-q')/2

Repulsive force F' = 1.3 N

We know F' = KQQ/r2 = KQ2/r2

From this Q2 = F'*r2/K = 1.0436 x10 -10

                  Q = 1.022 x10 -5

         (q -q ') / 2 = 1.022 x10 -5

             q - q ' = 2.044 x10 -5          ---------( 2)

From the equation (q +q') 2 = (q-q ') 2 + 4q q'

                                       = (4.178x10-10) + 4(1.686 x10 -10  )

                                       = 1.092 x10 -9

                             q + q ' = 3.305 x10 -5              ------( 3)

From eq (2) and eq (3),

q = 2.6745 x10 -5 C and q' = 0.6305 x10 -5C

but q' is negative, so q' = -0.6305 x10 -5C

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