A solid conducting sphere of radius R1𝑅1 has a charge of Q = 2.731 μC distributed evenly on its surface. A second solid conducting sphere of radius R2 = 0.7873 m𝑅2 = 0.7873 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 then 8.40E-7 C. What is the radius of the first sphere? m
Question
A solid conducting sphere of radius R1𝑅1 has a charge of Q = 2.731 μC distributed evenly on its surface. A second solid conducting sphere of radius R2 = 0.7873 m𝑅2 = 0.7873 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 then 8.40E-7 C. What is the radius of the first sphere? m
Solution
The problem involves the principle of conservation of charge and the property of conductors that the electric field inside a conductor in electrostatic equilibrium is zero.
When the two spheres are connected by a wire, charge will flow until the potential on both spheres is the same. The potential V of a sphere with charge Q and radius R is given by V = kQ/R, where k is Coulomb's constant (8.99 x 10^9 N m^2/C^2).
Let's denote the charge on the first sphere after disconnection as Q1 and the charge on the second sphere as Q2. We know that Q2 = 8.40E-7 C and the total charge Q = Q1 + Q2 = 2.731 μC = 2.731 x 10^-6 C.
Since the potentials on the two spheres are the same after disconnection, we have:
V1 = V2 kQ1/R1 = kQ2/R2 Q1/R1 = Q2/R2 Q1 = Q2 * R1/R2
Substituting Q1 into the equation for total charge gives:
Q = Q1 + Q2 = Q2 * R1/R2 + Q2 2.731 x 10^-6 C = 8.40E-7 C * R1/0.7873 m + 8.40E-7 C 2.731 x 10^-6 C - 8.40E-7 C = 8.40E-7 C * R1/0.7873 m 1.891 x 10^-6 C = 8.40E-7 C * R1/0.7873 m
Solving for R1 gives:
R1 = 1.891 x 10^-6 C * 0.7873 m / 8.40E-7 C R1 = 1.78 m
So, the radius of the first sphere is approximately 1.78 meters.
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