Final answer:
The magnitude of the EMF induced in the shrinking loop after 8.00s is approximately 14665.88 V.
Step-by-step explanation:
The emf (electromotive force) induced in a loop can be calculated using the formula EMF = -dΦ/dt, where dΦ/dt represents the rate of change of magnetic flux through the loop. In this case, the circumference of the loop is decreasing at a rate of 14.0 cm/s. Since the loop is circular, its circumference is equal to 2πr, where r is the radius of the loop. We can use this information to find the rate of change of flux, and hence the induced emf.
Given that the initial circumference of the loop is 162 cm, we can find the initial radius using the formula r = C/(2π), where C is the initial circumference. Substituting the values, we get r = 162/(2π) ≈ 25.79 cm.
Now, let's find the rate of change of flux using the formula dΦ/dt = B * A * dA/dt, where B is the magnetic field strength, A is the area of the loop, and dA/dt is the rate of change of the area. As per the given question, the loop is in a constant uniform magnetic field of magnitude 0.500 T, and the area of the loop can be calculated using the formula A = πr^2.
Substituting the values, we get A = π(25.79)^2 ≈ 2090.84 cm². Since the circumference is decreasing at a constant rate, the area of the loop is also decreasing at the same rate. Hence, dA/dt = -14.0 cm²/s.
By substituting the values in the formula, we can calculate the rate of change of flux as dΦ/dt = (0.500 T)(2090.84 cm²)(-14.0 cm²/s) ≈ - 14665.88 T cm²/s.
Finally, substituting the rate of change of flux into the formula for EMF, we get EMF = (-dΦ/dt) ≈ -(-14665.88 T cm²/s) ≈ 14665.88 V.