Final answer:
To keep the induced emf at zero, the rate at which the loop's radius increases, dr/dt, must counterbalance the rate at which the magnetic field decreases, ensuring that the change in loop area compensates for the change in magnetic field strength.
Step-by-step explanation:
The student's question revolves around the concept of electromagnetic induction, specifically within the context of a circular wire loop in a changing magnetic field. According to Faraday's Law of electromagnetic induction, the induced electromotive force (emf) in a loop is proportional to the rate of change of the magnetic flux through the loop. When the magnetic field decreases at a constant rate, it induces an emf in the loop which can be offset by changing the loop's radius.
To ensure that the induced emf is zero, we must adjust the rate of change of the loop's area to counterbalance the rate of change of the magnetic field. The formula for magnetic flux Φ is Φ = B ⋅ A, where B is the magnetic field strength and A is the area of the loop. The emf (ε) induced in a single loop is given by ε = -dΦ/dt. Substituting Φ with B ⋅ A (where A = πr² for a circular loop), and given that dB/dt is -1.6×10⁻² T/s, we can express ε as ε = -d(B ⋅ πr²)/dt.
To maintain ε = 0, we can differentiate the expression for Φ with respect to time and solve for dr/dt, the rate of change of the loop's radius.
Ultimately, the rate at which r should increase to maintain an induced emf of zero can be found using these relationships, ensuring the rate of change of the area, d(πr²)/dt, equals the rate at which the magnetic field strength is decreasing.