Question

Two small positively charged spheres have a combined charge of `5.0×10^-5 C`. If each sphere is repelled from the other by an electrostatic force of 1.0N when the spheres are 2.0m apart, what is the charge on each sphere?

Answer 1

`q_1 = 3 .84×10^-5 C`
`q_2 = 1 .16×10^-5 C`

Explanation:

We are are not given the values of the individual charges; let them be q1 and q2. The condition on the combined charge of the spheres gives us: `q_1 + q_2 =5.0×10^-5 C`
The next condition concerns the electrostatic force, and so it involves Coulomb’s Law. Both charges are positive because their sum is positive and they repel each other, thus
`|q1| = q1` and `|q2| = q2`
Now `F = k (q_1q_2)/r^2 =1 .0N` We know k and r, so we can solve for the value of the product of the charges:
`q_1q_2 = (1.0N)r^2/k`
#(1.0N)(2.0m)^2 8.99×10^9 N·m^2 C^ 2 =4 .449×10^-10 C^2#
Now we have two equations for the two unknowns q1 and q2.
`color(red)(q_2 =5.0×10^-5 −q_1)`
`q_1color(red)(q_2) = 4.449xx10^-10`
`q_1color(red)((5.0×10^-5 −q_1)) = 4.449xx10^-10`
`(5.0×10^-5q_1 −q_1^2) = 4.449xx10^-10`
`q_1^2 - (5.0×10^-5 C)q1 +4 .449×10^-10=0`
Use a quadratic formula
`q_(1,2) = ((5×10^-5) +- sqrt((5×10^-5)^2 - 4(4 .449×10^-10 )))/2`
`q_1 = 3 .84×10^-5 C; q_2 = 1 .16×10^-5 C`