Question

A technician wearing a circular metal band on his wrist moves his hand (from a region where the magnetic field is zero) into a uniform magnetic field of magnitude 2.0 T in a time of 0.48 s. If the diameter of the band is 5.5 cm and the field is at an angle of 45∘ with the plane of the metal band while the hand is in the field, find the magnitude of the average emf induced in the band

Answer 1

Faraday's law of electromagnetic induction is used to calculate the average emf induced in a metal band by considering the change in magnetic flux through the area of the band when it is moved into a uniform magnetic field at a given angle.

Step-by-step explanation:

To find the magnitude of the average emf induced in the technician's metal band, we can use Faraday's law of electromagnetic induction. The formula for the emf (ε) induced in a circular loop is given by ε = -dΦ/dt, where Φ is the magnetic flux through the loop and t is time. The magnetic flux is given by Φ = B * A * cos(θ), where B is the magnetic field strength, A is the area of the loop, and θ is the angle between the field and the normal to the plane of the loop.

In this scenario, the diameter of the band is 5.5 cm (which gives us a radius r of 2.75 cm or 0.0275 m), the magnetic field B is 2.0 T, the angle θ is 45°, and the time interval t is 0.48 s. The area A of the band is πr². Plugging in these values and considering the cosine of 45° (√2/2), the change in flux over the given time yields the average emf.

Φ_initial = 0 (since the initial magnetic field is zero), and Φ_final = B * A * cos(θ). The change in flux (∆Φ) is then Φ_final - Φ_initial. Dividing this by the time t gives the average emf induced in the band.

Answer 2

The magnitude of the average emf induced in the metal band is 0.067 V.

Step-by-step explanation:

To find the average emf induced in the metal band, we can use Faraday's law of electromagnetic induction. The formula for average emf induced is given by Emf = (B * A * cos(theta))/(t), where B is the magnetic field strength, A is the area, theta is the angle between the magnetic field and the plane of the band, and t is the time taken to move the hand into the field.

First, we calculate the area of the circular band. Since the diameter is given as 5.5 cm, the radius will be half of that, which is 2.75 cm or 0.0275 m. The area of the circle is A = pi * r^2. Therefore, A = pi * (0.0275^2) = 0.0024 m^2.

Using the given values, the formula becomes Emf = (2.0 T * 0.0024 m^2 * cos(45∘))/(0.48 s). Evaluating this expression gives Emf = 0.067 V (to three significant figures).