Note: Figure not drawn to scale.
1998E1. The small sphere A in the diagram above has a charge of 120μC. The large sphere B1 is a thin shell of nonconducting material with a net charge that is uniformly distributed over its surface. Sphere B1 has a mass of 0.025 kg , a radius of 0.05 m , and is suspended from an uncharged, nonconducting thread. Sphere B1 is in equilibrium when the thread makes an angle θ=20∘ with the vertical. The centers of the spheres are at the same vertical height and are a horizontal distance of 1.5 m apart, as shown.
a. Calculate the charge on sphere B1.
b. Suppose that sphere B1 is replaced by a second suspended sphere B2 that has the same mass, radius, and charge, but that is conducting. Equilibrium is again established when sphere A is 1.5 m from sphere B2 and their centers are at the same vertical height. State whether the equilibrium angle θ2 will be less than, equal to, or greater than 20∘. Justify your answer.
The sphere B2 is now replaced by a very long, horizontal, nonconducting tube, as shown in the top view below. The tube is hollow with thin walls of radius R=0.20m and a uniform positive charge per unit length of λ=+0.10μC/m.
Top View
c. Use Gauss's law to show that the electric field at a perpendicular distance r from the tube is given by the expression E=(1.8×103)/rN/C, where r>R and r is in meters.
d. The small sphere A with charge 120μC is now brought into the vicinity of the tube and is held at a distance of r=1.5m from the center of the tube. Calculate the repulsive force that the tube exerts on the sphere.
e. Calculate the work done against the electrostatic repulsion to move sphere A toward the tube from a distance r=1.5m to a distance r=0.3m from the tube.