Let's break down the problem step by step: ### **Given Data** - **Magnetic field, \( B \) = 2. T** - **Time, \( \Delta t \) = .48 s** - **Diameter of band, \( d \) = 5.5 cm = .055 m** - **Angle, \( \theta \) = 45°** - **Initial magnetic field = T** ### **Step 1: Find Area of the Band (\( A \))** The band is circular, so: \[ A = \pi r^2 = \pi \left(\frac{d}{2}\right)^2 = \pi \left(\frac{.055}{2}\right)^2 = \pi (.0275)^2 \approx .002376 \text{ m}^2 \] ### **Step 2: Find Change in Magnetic Flux (\( \Delta\Phi \))** Magnetic flux, \( \Phi = B A \cos \theta \) - **Initial flux** (\( \Phi_i \)) = (since initial B = ) - **Final flux** (\( \Phi_f \)) = \( B A \cos \theta \) So, \[ \Delta \Phi = \Phi_f - \Phi_i = B A \cos \theta - = B A \cos \theta \] ### **Step 3: Use Faraday’s Law for Average EMF** \[ \text{Average emf} = \left| \frac{\Delta \Phi}{\Delta t} \right| \] \[ \text{Average emf} = \frac{B A \cos \theta}{\Delta t} \] ### **Step 4: Plug In Values** \[ \cos 45^\circ = \frac{\sqrt{2}}{2} \approx .707 \] \[ \text{Average emf} = \frac{2. \times .002376 \times .707}{.48} \] Calculate numerator: \[ 2. \times .002376 = .004752 \] \[ .004752 \times .707 \approx .00336 \] Now divide by \( \Delta t \): \[ \text{Average emf} = \frac{.00336}{.48} \approx .007 \text{ V} \] Convert to mV: \[ .007 \text{ V} = 7. \text{ mV} \] --- ## **Final Answer** \[ \boxed{7. \, \text{mV}} \] --- **Summary Table** | Quantity | Value | |-------------|------------| | Area (\(A\))| .002376 m²| | Angle (\(\theta\))| 45° | | Magnetic Field (\(B\))| 2. T | | Time (\(\Delta t\))| .48 s | | Average emf | 7. mV |