27. ssm A magnetic field is passing through a loop of wire whose area is 0.018 m². The direction of the magnetic field is parallel to the normal to the loop, and the magnitude of the field is increasing at the rate of 0.20 T/s. (a) Determine the magnitude of the emf induced in the loop. (b) Suppose that the area of the loop can be enlarged or shrunk. If the magnetic field is increasing as in part (a), at what rate (in m²/s) should the area be changed at the instant when B = 1.8 T if the induced emf is to be zero? Explain whether the area is to be enlarged or shrunk.
27. ssm A magnetic field is passing through a loop of wire whose area is 0.018 m². The direction of the magnetic field is parallel to the normal to the loop, and the magnitude of the field is increasing at the rate of 0.20 T/s. (a) Determine the magnitude of the emf induced in the loop. (b) Suppose that the area of the loop can be enlarged or shrunk. If the magnetic field is increasing as in part (a), at what rate (in m²/s) should the area be changed at the instant when B = 1.8 T if the induced emf is to be zero? Explain whether the area is to be enlarged or shrunk.
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![### Electromagnetic Induction Problem
#### Problem 27
**Context:**
A magnetic field is passing through a loop of wire whose area is 0.018 m². The direction of the magnetic field is parallel to the normal to the loop, and the magnitude of the field is increasing at the rate of 0.20 T/s.
**Tasks:**
**(a) Determine the magnitude of the emf induced in the loop.**
**(b) Suppose that the area of the loop can be enlarged or shrunk. If the magnetic field is increasing as in part (a), at what rate (in m²/s) should the area be changed at the instant when \( B = 1.8\, \text{T} \) if the induced emf is to be zero? Explain whether the area is to be enlarged or shrunk.**
### Explanation and Solution:
**(a) Induced Emf Calculation:**
The electromotive force (emf) induced in a loop is given by Faraday's law of electromagnetic induction:
\[ \mathcal{E} = -\frac{d\Phi_B}{dt} \]
Where:
\[ \Phi_B = B \cdot A \]
is the magnetic flux, with \( B \) being the magnetic field and \( A \) being the area.
Given:
- \( A = 0.018\, \text{m}² \)
- \(\frac{dB}{dt} = 0.20\, \text{T/s} \)
Substitute these values into the expression for \(\mathcal{E}\):
\[ \mathcal{E} = -A \cdot \frac{dB}{dt} \]
\[ \mathcal{E} = -0.018\, \text{m}² \cdot 0.20\, \text{T/s} \]
\[ \mathcal{E} = -0.0036\, \text{V} \]
The magnitude of the induced emf is:
\[ \mathcal{E} = 0.0036\, \text{V} \]
**(b) Changing Area to Induce Zero Emf:**
To have the induced emf be zero, the change in the magnetic flux (\(\Phi_B\)) must be zero. This can be due to changes in both \(B\) and \(A\) compensating each other.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc02bfde4-a78d-4551-93f6-03d589a5c8dc%2F51d1e1fd-b9b7-4753-b17c-bd3fd7c9f956%2Fmtts19f_processed.png&w=3840&q=75)
Transcribed Image Text:### Electromagnetic Induction Problem
#### Problem 27
**Context:**
A magnetic field is passing through a loop of wire whose area is 0.018 m². The direction of the magnetic field is parallel to the normal to the loop, and the magnitude of the field is increasing at the rate of 0.20 T/s.
**Tasks:**
**(a) Determine the magnitude of the emf induced in the loop.**
**(b) Suppose that the area of the loop can be enlarged or shrunk. If the magnetic field is increasing as in part (a), at what rate (in m²/s) should the area be changed at the instant when \( B = 1.8\, \text{T} \) if the induced emf is to be zero? Explain whether the area is to be enlarged or shrunk.**
### Explanation and Solution:
**(a) Induced Emf Calculation:**
The electromotive force (emf) induced in a loop is given by Faraday's law of electromagnetic induction:
\[ \mathcal{E} = -\frac{d\Phi_B}{dt} \]
Where:
\[ \Phi_B = B \cdot A \]
is the magnetic flux, with \( B \) being the magnetic field and \( A \) being the area.
Given:
- \( A = 0.018\, \text{m}² \)
- \(\frac{dB}{dt} = 0.20\, \text{T/s} \)
Substitute these values into the expression for \(\mathcal{E}\):
\[ \mathcal{E} = -A \cdot \frac{dB}{dt} \]
\[ \mathcal{E} = -0.018\, \text{m}² \cdot 0.20\, \text{T/s} \]
\[ \mathcal{E} = -0.0036\, \text{V} \]
The magnitude of the induced emf is:
\[ \mathcal{E} = 0.0036\, \text{V} \]
**(b) Changing Area to Induce Zero Emf:**
To have the induced emf be zero, the change in the magnetic flux (\(\Phi_B\)) must be zero. This can be due to changes in both \(B\) and \(A\) compensating each other.
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