Untitled Note

Alright, let's delve into the solution process for each part of the problem: ### Part A: You were asked to find the total magnetic flux \( \Phi_B \) through the ring as a function of time. 1. **Magnetic Flux Calculation**: \[ \Phi_B = B \times A \] Where \( A \) is the area of the circle, which is \( A = \pi r_0^2 \). Given the magnetic field \( B \) as: \[ B = B_0 \left[ 1 - 3 \left( \frac{t}{t_0} \right)^2 2 \left( \frac{t}{t_0} \right)^3 \right] \hat{k} \] To find \( \Phi_B \), you would multiply the magnetic field \( B \) by the area \( A \): \[ \Phi_B = B_0 \left[ 1 - 3 \left( \frac{t}{t_0} \right)^2 2 \left( \frac{t}{t_0} \right)^3 \right] \times \pi r_0^2 \] ### Part B: Determine the emf induced in the ring at \( t = 5.00 \times 10^{-3} s \). Using Faraday's Law: \[ \epsilon = - \frac{d\Phi_B}{dt} \] 2. **Differentiation**: Differentiate the expression for \( \Phi_B \) with respect to time \( t \). After differentiating, plug in the value \( t = 5.00 \times 10^{-3} s \) to get the induced emf at that specific time. The calculated result for this specific time was \( |\epsilon| = 6.65 \times 10^{-2} \) V. ### Part C: For the polarity of the emf induced, you'd use Lenz's Law. 3. **Lenz's Law Application**: To determine the direction of the induced emf, examine the rate of change of the magnetic field. If the magnetic field is increasing, the induced emf (and current) will act in a direction to oppose this increase, and vice versa. By looking at the given magnetic field equation and its derivatives, you'd check the sign of the rate of change at \( t = 5.00 \times 10^{-3} s \). The sign would tell you whether the magnetic field is increasing or decreasing, and thus you'd know the direction of the induced emf. --- The given answers in the image suggest that these steps were correctly followed to arrive at the correct solutions for each part of the problem.