Answer:
The average value of the voltage induced in the coil is \( 21.6 \, \text{V} \).
Explanation:
To solve this problem, we'll use Faraday's law of electromagnetic induction, which states that the induced electromotive force (emf) in a closed circuit is equal to the rate of change of magnetic flux through the circuit. The formula for Faraday's law is:
\[ \text{emf} = -\frac{d\Phi}{dt} \]
Where:
- \( \text{emf} \) is the induced electromotive force,
- \( \frac{d\Phi}{dt} \) is the rate of change of magnetic flux through the coil.
(a) To find the initial magnetic flux through the coil, we'll use the formula for magnetic flux:
\[ \Phi = B \cdot A \cdot N \]
Where:
- \( \Phi \) is the magnetic flux,
- \( B \) is the magnetic field strength (2.4 T),
- \( A \) is the cross-sectional area of the coil (0.06 m²),
- \( N \) is the number of turns in the coil (60 turns).
Substituting the given values:
\[ \Phi = (2.4 \, \text{T}) \times (0.06 \, \text{m²}) \times (60) \]
\[ \Phi = 8.64 \, \text{Wb} \]
So, the initial magnetic flux through the coil is \( 8.64 \, \text{Wb} \).
(b) Now, to find the average value of the voltage induced in the coil, we'll use Faraday's law:
\[ \text{emf} = -\frac{d\Phi}{dt} \]
Since the coil is rapidly removed from the magnetic field, we can assume a constant rate of change of magnetic flux. Therefore, the average value of the induced voltage is:
\[ \text{Average emf} = \frac{\Delta \Phi}{\Delta t} \]
Where \( \Delta \Phi \) is the change in magnetic flux and \( \Delta t \) is the change in time.
Given that the time \( \Delta t \) is 0.4 s, we can calculate \( \Delta \Phi \) as the difference between the initial and final magnetic flux:
\[ \Delta \Phi = \Phi_{\text{initial}} - \Phi_{\text{final}} \]
\[ \Delta \Phi = 8.64 \, \text{Wb} - 0 \, \text{Wb} \]
\[ \Delta \Phi = 8.64 \, \text{Wb} \]
Now, we can calculate the average emf:
\[ \text{Average emf} = \frac{8.64 \, \text{Wb}}{0.4 \, \text{s}} \]
\[ \text{Average emf} = 21.6 \, \text{V} \]
So, the average value of the voltage induced in the coil is \( 21.6 \, \text{V} \).