Each of two small non-conducting spheres is charged positively,the combined charge being 40 $\mu$C.When the two spheres are 50 cm apart,each sphere is repelled from the other by a force of magnitude 2.0 N.Determine the magnitude of the smaller of the two charges.
A)1.4 $\mu$C
B)1.1 $\mu$C
C)2.0 $\mu$C
D)3.3 $\mu$C
E)17 $\mu$C

Question

Quizplus · Accepted Answer

The force between the two spheres is given by Coulomb's law:$$F = k\frac{q_1q_2}{d^2}$$where k is Coulomb's constant, q1 and q2 are the charges on the spheres, and d is the distance between them.We are given that the force between the spheres is 2.0 N, the distance between them is 50 cm, and the combined charge on the spheres is 40 μC. We can use this information to solve for the magnitude of the smaller charge.First, we need to find the total charge on the spheres. We are given that the combined charge is 40 μC, so the total charge is:$$q_1 + q_2 = 40 \mu C$$Next, we need to find the distance between the spheres in meters. We are given that the distance is 50 cm, so the distance in meters is:$$d = 50 cm = 0.5 m$$Now we can substitute these values into Coulomb's law and solve for the magnitude of the smaller charge:$$2.0 N = k\frac{q_1q_2}{d^2}$$$$2.0 N = (9 	imes 10^9 Nm^2/C^2)\frac{q_1q_2}{(0.5 m)^2}$$$$q_1q_2 = 1.11 	imes 10^{-9} C^2$$We also know that the total charge on the spheres is 40 μC, so we can write:$$q_1 + q_2 = 40 \mu C$$$$q_1 = 40 \mu C - q_2$$Substituting this into the equation above, we get:$$(40 \mu C - q_2)q_2 = 1.11 	imes 10^{-9} C^2$$$$40 \mu C q_2 - q_2^2 = 1.11 	imes 10^{-9} C^2$$$$q_2^2 - 40 \mu C q_2 + 1.11 	imes 10^{-9} C^2 = 0$$Using the quadratic formula, we can solve for q2:$$q_2 = \frac{40 \mu C \pm \sqrt{(40 \mu C)^2 - 4(1.11 	imes 10^{-9} C^2)}}{2}$$$$q_2 = \frac{40 \mu C \pm 39.99 \mu C}{2}$$$$q_2 = 1.0 \mu C 	ext{ or } q_2 = 40 \mu C$$The smaller charge is 1.0 μC, so the answer is A.

Each of Two Small Non-Conducting Spheres Is Charged Positively,the Combined μ\muμ

Each of Two Small Non-Conducting Spheres Is Charged Positively,the Combined $\mu$