Certainly! Let's solve the problem step-by-step, detailing the calculations and concepts involved.
### Given Data:
1. Area of the loop, [tex]\( A \)[/tex]: [tex]\( 9.06 \times 10^{-2} \ \text{m}^2 \)[/tex]
2. Initial magnetic field, [tex]\( B_i \)[/tex]: [tex]\( 3.68 \ \text{T} \)[/tex]
3. Rate at which the magnetic field is decreasing, [tex]\( \frac{dB}{dt} \)[/tex]: [tex]\( -0.196 \ \text{T/s} \)[/tex]
### Find:
The induced electromotive force (emf) in the loop.
### Concepts and Formulae:
Faraday's Law of Induction states that the induced emf in a loop is given by:
[tex]\[
\text{emf} = - \frac{d\Phi}{dt}
\][/tex]
where [tex]\( \Phi \)[/tex] is the magnetic flux. Magnetic flux [tex]\( \Phi \)[/tex] through a single loop is defined as:
[tex]\[
\Phi = B \cdot A
\][/tex]
Since the magnetic field [tex]\( B \)[/tex] is perpendicular to the area [tex]\( A \)[/tex], the flux simplifies to:
[tex]\[
\Phi = B \cdot A
\][/tex]
When the magnetic field is changing with time, the change in magnetic flux can be expressed as:
[tex]\[
\frac{d\Phi}{dt} = A \cdot \frac{dB}{dt}
\][/tex]
So, the induced emf becomes:
[tex]\[
\text{emf} = - A \cdot \frac{dB}{dt}
\][/tex]
### Step-by-Step Solution:
1. Identify the area of the loop:
[tex]\[
A = 9.06 \times 10^{-2} \ \text{m}^2
\][/tex]
2. Identify the rate of change of the magnetic field:
[tex]\[
\frac{dB}{dt} = -0.196 \ \text{T/s} \ (\text{the field is decreasing, hence the negative sign})
\][/tex]
3. Substitute the known values into the formula:
[tex]\[
\text{emf} = - A \cdot \frac{dB}{dt}
\][/tex]
4. Calculate the induced emf:
[tex]\[
\text{emf} = - (9.06 \times 10^{-2} \ \text{m}^2) \cdot (-0.196 \ \text{T/s})
\][/tex]
[tex]\[
\text{emf} = (9.06 \times 10^{-2}) \times 0.196 \ \text{V}
\][/tex]
[tex]\[
\text{emf} = 0.0177576 \ \text{V}
\][/tex]
Therefore, the induced electromotive force (emf) in the loop is approximately [tex]\( -0.0177576 \ \text{V} \)[/tex]. The negative sign indicates the direction of the induced emf as per Lenz's Law, which states that the induced emf will act in such a way as to oppose the change in magnetic flux.
