Question

To apply Ampère's law to find the magnetic field inside an infinite solenoid.

In this problem we will apply Ampère's law, written

∮B⃗ (r⃗ )⋅dl⃗ =μ0Iencl,

to calculate the magnetic field inside a very long solenoid (only a relatively short segment of the solenoid is shown in the pictures). The solenoid has length L, diameter D, and n turns per unit length with each carrying current I. (Figure 1) It is usual to assume that the component of the current along the z axis is negligible. (This may be assured by winding two layers of closely spaced wires that spiral in opposite directions.)

From symmetry considerations it is possible to show that far from the ends of the solenoid, the magnetic field is axial.

Part A)

Assume that loop B (in the Part A figure) has length L along k^ (the z direction). What is the loop integral in Ampère's law? Assume that the top end of the loop is very far from the solenoid (even though it may not look like it in the figure), so that the field there is assumed to be small and can be ignored.

Express your answer in terms of Bin, L, and other quantities given in the introduction.

Part B)

Find Bin, the z component of the magnetic field inside the solenoid where Ampère's law applies.

Express your answer in terms of L, D, n, I, and physical constants such as μ0.

Answer 1