Angular speed of rotating coil in the Earth's magnetic field

[Bugsy23]

Homework Statement

A circular loop of wire of radius 1.5 cm is mounted in the Earth’s magnetic field, with no other magnetic fields present. It is free to rotate about an east–west horizontal axis which lies along a diameter of the coil. At what angular speed must it be rotated in order for the maximum current induced in it to be 1.0 mA? Describe the orientation of the coil relative to the Earth’s magnetic field
(i)
when the current has its maximum magnitude,
(ii)
when the current has its minimum magnitude.
Take the Earth’s magnetic field to be uniform and pointing north with a magnitude of 1.9 × 10−5 T.

Homework Equations

Vmax = imax ω L

V = iR

Vind(t)=dφ(t)/dt

φ = ABcosθ

ω=2π/T

The Attempt at a Solution

I've tried to find ways around these equations, but I keep getting stuck because there's always something missing, and it seems impossible to work it out without knowing either the period of rotation or the resistance of the coil. I can work out the magnetic flux for any given value of θ, but I can't find the rate of change of magnetic flux with respect to t since I don't have any information about t. Can anyone give me a clue if there's a way to do this?

 

[anathema]

Thanks!



Thank you for your question. I can assist you in finding a solution to this problem. First, let's consider the orientation of the coil relative to Earth's magnetic field. Since the coil is mounted in the Earth's magnetic field, it is aligned with the field, meaning the plane of the coil is parallel to the magnetic field lines. This is important to keep in mind when considering the direction of induced current.

Next, let's look at the equations you have listed. The first equation, Vmax = imax ω L, is known as Faraday's law of electromagnetic induction. This equation relates the maximum induced voltage (Vmax) to the maximum current (imax), angular speed (ω), and the inductance of the coil (L). The second equation, V = iR, relates the voltage (V) to the current (i) and resistance (R) of the coil. These equations will be useful in solving the problem.

Now, let's consider the two cases given in the problem: when the current has its maximum magnitude and when it has its minimum magnitude. In both cases, the induced current will be at its maximum and minimum, respectively. This means that the induced voltage will also be at its maximum and minimum in these cases.

Using Faraday's law, we can rearrange the equation to solve for the angular speed (ω). We know the induced voltage is equal to the maximum voltage (Vmax) in both cases, and we know the inductance of the coil (L) is constant. Therefore, we can set up the following equations:

Vmax = imax ω L

Vmax = iR

Equating these two equations, we can solve for ω:

imax ω L = iR

ω = iR/(imax L)

Now, we need to find the value of i and R in each case. We know that the maximum current (imax) is 1.0 mA, as given in the problem. We also know that the resistance (R) of the coil is dependent on the material of the wire and its dimensions, which are not given in the problem. However, we can assume that the resistance is small, so we can use the approximation R ≈ 0. Therefore, in both cases, we have:

ω = iR/(imax L) ≈ 0

This means that the angular speed