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?

1 Answer
Mar 9, 2016

Answer:

#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 #