A solid conducting sphere of radius R1 = 1.466 m𝑅1 = 1.466 m has a charge of Q distributed evenly on its surface. A second solid conducting sphere of radius R2 = 0.7079 m𝑅2 = 0.7079 m is initially uncharged. The second sphere is located a distance of 10 m from the first sphere. The two spheres are momentarily connected with a wire and then the wire is disconnected. The charge on the second sphere is now 3.08E-6 C. What was the original charge Q on the first sphere before the two spheres were connected? C
Question
A solid conducting sphere of radius R1 = 1.466 m𝑅1 = 1.466 m has a charge of Q distributed evenly on its surface. A second solid conducting sphere of radius R2 = 0.7079 m𝑅2 = 0.7079 m is initially uncharged. The second sphere is located a distance of 10 m from the first sphere. The two spheres are momentarily connected with a wire and then the wire is disconnected. The charge on the second sphere is now 3.08E-6 C. What was the original charge Q on the first sphere before the two spheres were connected? C
Solution
The charge on a conductor is distributed evenly on its surface. When two conductors are connected, the charge will redistribute so that the potential on both conductors is the same.
The potential V of a charged sphere is given by V = kQ/R, where k is Coulomb's constant, Q is the charge, and R is the radius of the sphere.
After the two spheres are connected, the potentials must be equal, so we have:
V1 = V2 kQ1/R1 = kQ2/R2
We can cancel out the k's and rearrange to find Q1:
Q1 = Q2 * R1/R2
We know that Q2 = 3.08E-6 C, R1 = 1.466 m, and R2 = 0.7079 m. Plugging these values in gives:
Q1 = 3.08E-6 C * 1.466 m / 0.7079 m = 6.38E-6 C
However, this is the charge on the first sphere after the wire is disconnected. The original charge Q on the first sphere before the two spheres were connected is the sum of the charges on the two spheres after the wire is disconnected, so:
Q = Q1 + Q2 = 6.38E-6 C + 3.08E-6 C = 9.46E-6 C
So, the original charge Q on the first sphere before the two spheres were connected was 9.46E-6 C.
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