What do you expect to observe as the cylinder falls through

Question

Question 1.

Consider a long straight conducting wire that terminates in a metal sphere of radius r = 0.50 ± 0.01 cm. Suppose the wire is carrying a current of / = 3.0 ± 0.2 A in order to charge the sphere (positively).

(a) Sketch the shape of the electric and magnetic fields around the wire and the sphere.

(b) Discuss the relative merits of using the Biot-Savart law and Ampere's law to calculate the magnetic field around the wire. In this case, is it necessary to include the Maxwell correction to Ampere's law?

(c) Calculate the magnetic field strength at point P. which lies a distance of d = 1.00 ± 0.01 cm from the wire and I? = 10.0 ± 0.1 cm from the centre of the ball.

(d) Calculate the uncertainty in the magnetic field strength. Does the fact that this wire is not infinite in extent make a significant impact on the results? Explain.

Question 2.

A coil of wire has a radius of r₁, a resistance of R₁, and contains N₁ turns of wire. It is connected to a voltage source V₁. A second, smaller ring has a radius of r₂, a resistance of R2, and AT₂ turns of wire. Ring 2 is placed in the middle of the Ring 1 such that their centres are concentric as shown in the figure below.

When the switch is initially closed:

(a) What is the strength of the magnetic field E₁ generated by the current in Ring 1?

(b) What is the magnetic flux passing through Ring 2 due to A?

(c) Derive an expression for the induced emf in Ring 2, 6₂? How much current flows in Ring 2 and in which direction?

(d) The current flowing in Ring 2 produces its own magnetic field, which in turn induces an emf in Ring 1, e_l. How much is this? In which direction does this current flow?

Ring 2 is now removed. A solid metal cylinder with the same radius as Ring 2 (and height=radius) is dropped through the middle of Ring 1.

(e) What do you expect to observe as the cylinder falls through the ring? Include times before, during and after the cylinder passes through the ring. Justify your answer and list any assumptions you have to make. Hint: You may want to use some diagrams.

(f) Would you expect the same result if Ring 2 were dropped through the loop? Justify your answer.