Final answer:
The question deals with calculating the induced emf in a rectangular loop due to a long straight wire carrying constant current. Using Faraday's Law and the magnetic field from a straight wire, we differentiate the flux concerning time to find the magnitude of the emf as the loop changes width with respect to time.
Step-by-step explanation:
The student's question pertains to the electromagnetic induction phenomenon, where a changing magnetic field induces an electromotive force (emf) in a circuit. The question involves calculating the magnitude of the induced emf in a rectangular loop that is changing width near a long straight wire carrying a constant current.
According to Faraday's Law of Electromagnetic Induction, the emf (ε) induced in a loop is given by the rate of change of magnetic flux through the loop. The magnetic flux (Φ) is the product of the magnetic field (B) due to the long straight wire, the area of the loop (A), and the cosine of the angle (θ) between the magnetic field and normal to the loop's surface. Since the loop's plane and the wire lie in the same plane, θ = 0, making cos(θ) = 1. The magnetic field produced by a long straight wire at a distance x is given by B = (μ0 * I) / (2π * x), where μ0 is the permeability of free space and I is the current in the wire. The area of the loop is A = a * b, with given dimensions a and b.
To find the induced emf, one must differentiate the flux with respect to time. The changing variable here is x, which changes at a rate dx/dt. Therefore, we get:
ε = -dΦ/dt = -d/dt(B * a * b) = -d/dt((μ0 * I * a * b) / (2π * x)) = -(μ0 * I * a * b) / (2π) * d(1/x)/dt.
Plugging in the values for a constant rate of change of x (dx/dt), the current I, the dimensions a and b, and μ0 (which is approximately 4π * 10^-7 T*m/A), we can compute the magnitude of the induced emf at the instant when x = 6.0 cm.