To solve this problem, we'll approach it step-by-step.
### Step 1: Calculate the Magnetic Flux through the Coil
The magnetic flux [tex]\(\Phi_B\)[/tex] through a coil is given by the formula:
[tex]\[
\Phi_B = N \cdot B \cdot A
\][/tex]
where:
- [tex]\(N\)[/tex] is the number of turns in the coil,
- [tex]\(B\)[/tex] is the magnetic field strength,
- [tex]\(A\)[/tex] is the area of the coil.
Given:
- [tex]\(N = 220\)[/tex] turns,
- [tex]\(B_0 = 0.0605 \, T\)[/tex],
- Radius of the coil, [tex]\(r = 10.5 \, \text{cm} = 0.105 \, m\)[/tex].
First, we need to calculate the area [tex]\(A\)[/tex] of the circular coil:
[tex]\[
A = \pi r^2
\][/tex]
Substitute the radius:
[tex]\[
A = \pi \times (0.105 \, m)^2
\][/tex]
Next, we compute the magnetic flux [tex]\(\Phi_B\)[/tex]:
[tex]\[
\Phi_B = 220 \times 0.0605 \times \pi \times (0.105)^2
\][/tex]
The result of this calculation is:
[tex]\[
\Phi_B = 0.46100594536756356 \, T \cdot m^2
\][/tex]
### Step 2: Calculate the Induced EMF
The induced emf [tex]\(|E|\)[/tex] during a change in magnetic field is given by Faraday's Law of Induction:
[tex]\[
|E| = N \cdot \frac{\Delta B \cdot A}{\Delta t}
\][/tex]
where:
- [tex]\(\Delta B\)[/tex] is the change in magnetic field,
- [tex]\(\Delta t\)[/tex] is the time interval over which the change occurs.
Given:
- Final magnetic field, [tex]\(B_0 = 0.200 \, T\)[/tex],
- Initial magnetic field, [tex]\(B_0 = 0.0605 \, T\)[/tex],
- Time interval, [tex]\(\Delta t = 0.480 \, \text{s}\)[/tex].
Calculate the change in magnetic field:
[tex]\[
\Delta B = B_0 \, \text{final} - B_0 \, \text{initial} = 0.200 - 0.0605 = 0.1395 \, T
\][/tex]
Substitute all values into the emf formula:
[tex]\[
|E| = 220 \times \frac{0.1395 \times \pi \times (0.105)^2}{0.480}
\][/tex]
The result of this calculation is:
[tex]\[
|E| = 2.214543022685094 \, V
\][/tex]
### Summary
The magnitude of the magnetic flux [tex]\(\Phi_B\)[/tex] through the coil is:
[tex]\[
\Phi_B = 0.46100594536756356 \, T \cdot m^2
\][/tex]
The magnitude of the induced emf [tex]\(|E|\)[/tex] during the time interval is:
[tex]\[
|E| = 2.214543022685094 \, V
\][/tex]
