Question

circular wire loop of radius r = 20 cm is immersed in a uniform magnetic field B= 0.520 T with its plane normal to the direction of the field. If the field magnitude then decreases at a constant rate of −1.7×10−2 T/s, at what rate should r increase so that the induced emf within the loop is zero?

Answer 1

To ensure zero induced emf in the loop, the radius must increase at a rate given by r' = - (B' * r) / (2 * B). By substituting the given values for B, B', and r, we can calculate the required rate at which the radius of the loop must increase.

Step-by-step explanation:

The student's question pertains to the concept of electromagnetic induction in a circular loop of wire within a changing magnetic field, specifically applying Faraday's Law of Induction and Lenz's Law. We want to find at what rate the radius of the loop should increase to ensure that the induced emf (electromotive force) is zero when the magnetic field is decreasing at a constant rate.

Faraday's Law states that the induced emf in a loop is equal to the negative rate of change of the magnetic flux through the loop. Mathematically, ® = -dΦ/dt, where ® is the emf and Φ is the magnetic flux. Magnetic flux (Φ) is the product of the magnetic field (B), the area of the loop (A), and the cosine of the angle between the field and the normal to the loop's plane. In this case, the plane of the loop is normal to the magnetic field, so the angle is 0 degrees, and cos(0) = 1, simplifying the equation to Φ = B * A. Since the loop is circular, A = π * r^2, where r is the radius.

The rate of change of flux due to the changing magnetic field is therefore -B' * A, and the rate of change of flux due to the changing area (as the radius increases) is -B * 2π * r * r'. To have zero induced emf, these two rates should cancel each other out, hence B' * π * r^2 + B * 2π * r * r' = 0. Solving for r' gives r' = - (B' * r) / (2 * B). Substituting the given values B = 0.520 T, B' = -1.7×10⁻² T/s, and r = 0.20 m yields r' = - ((-1.7×10⁻² T/s) * 0.20 m) / (2 * 0.520 T), which calculates to a positive value, indicating that the radius must increase at that rate.

Answer 2

Therefore, the radius r should increase at a rate of approximately 0.0109 meters per second to maintain zero induced emf in the loop.

The key to solving this problem lies in Faraday's Law of Electromagnetic Induction, which states:

ε = -N dΦ/dt

where:

ε is the induced emf in the loop

N is the number of turns (1 in this case)

Φ is the magnetic flux through the loop

dΦ/dt is the rate of change of magnetic flux

For a circular loop in a uniform magnetic field, the magnetic flux is given by:

Φ = BA

where:

B is the magnetic field strength

A is the area of the loop

In this case, A = πr^2.

We are given that the magnetic field magnitude is decreasing at a constant rate of -1.7 × 10^-2 T/s. Therefore, the rate of change of magnetic flux is:

dΦ/dt = -Bπr^2 d(B)/dt = -1.7 × 10^-2 πr^2 (0.520 T/s)

We want the induced emf ε to be zero. From Faraday's Law, this means:

0 = -1.7 × 10^-2 πr^2 (0.520 T/s) - N r dr/dt

Since N = 1, we can solve for dr/dt:

dr/dt = (1.7 × 10^-2 πr^2 (0.520 T/s)) / r

Substituting r = 0.2 m (20 cm) and the given values:

dr/dt = (1.7 × 10^-2 π (0.2 m)^2 (0.520 T/s)) / 0.2 m

≈ 0.0109 m/s