The magnetic field inside a 5.0-cm-diameter solenoid is 2.0 T and decreasing at 4.0 T/s. What current flows in a 4.0 cm diameter circular wire loop of resistance R = 0.30 2 located at the center of the solenoid and aligned with the same axis?
The magnetic field inside a 5.0-cm-diameter solenoid is 2.0 T and decreasing at 4.0 T/s. What current flows in a 4.0 cm diameter circular wire loop of resistance R = 0.30 2 located at the center of the solenoid and aligned with the same axis?
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![**Magnetic Field and Induced Current Calculation**
A solenoid with a diameter of 5.0 cm has a magnetic field strength of 2.0 T, which is decreasing at a rate of 4.0 T/s. In the center of this solenoid and aligned with the same axis, there is a circular wire loop with a diameter of 4.0 cm and a resistance of R = 0.30 Ω.
The problem is to determine the current flowing in the wire loop due to the changing magnetic field.
**Concept Explanation:**
1. **Induced EMF:** According to Faraday's Law of electromagnetic induction, a changing magnetic field within a closed loop induces an electromotive force (EMF).
2. **Calculating Induced Current:** The induced EMF (\( \epsilon \)) can be calculated using Faraday's Law:
\[
\epsilon = -\frac{d\Phi_B}{dt}
\]
where \( \Phi_B \) is the magnetic flux. The current (\( I \)) can be determined using Ohm's law:
\[
I = \frac{\epsilon}{R}
\]
-- where \( R \) is the resistance of the loop.
3. **Magnetic Flux (\( \Phi_B \)):** This is the product of the magnetic field (\( B \)), the area (\( A \)) of the wire loop, and the cosine of the angle between the field and the normal to the area. For a perpendicular arrangement:
\[
\Phi_B = B \times A
\]
4. **Area of the Loop:** The area, \( A \), of the loop can be calculated as:
\[
A = \pi \times \left(\frac{d}{2}\right)^2
\]
where \( d \) is the diameter of the loop.
This problem applies these principles to find the induced current resulting from the decreasing magnetic field within the solenoid.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff80ba54f-18b4-4d71-8959-517d0920bb4d%2F61e74fd9-2321-415d-9156-d6bdb9f08bac%2F96m8irf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Magnetic Field and Induced Current Calculation**
A solenoid with a diameter of 5.0 cm has a magnetic field strength of 2.0 T, which is decreasing at a rate of 4.0 T/s. In the center of this solenoid and aligned with the same axis, there is a circular wire loop with a diameter of 4.0 cm and a resistance of R = 0.30 Ω.
The problem is to determine the current flowing in the wire loop due to the changing magnetic field.
**Concept Explanation:**
1. **Induced EMF:** According to Faraday's Law of electromagnetic induction, a changing magnetic field within a closed loop induces an electromotive force (EMF).
2. **Calculating Induced Current:** The induced EMF (\( \epsilon \)) can be calculated using Faraday's Law:
\[
\epsilon = -\frac{d\Phi_B}{dt}
\]
where \( \Phi_B \) is the magnetic flux. The current (\( I \)) can be determined using Ohm's law:
\[
I = \frac{\epsilon}{R}
\]
-- where \( R \) is the resistance of the loop.
3. **Magnetic Flux (\( \Phi_B \)):** This is the product of the magnetic field (\( B \)), the area (\( A \)) of the wire loop, and the cosine of the angle between the field and the normal to the area. For a perpendicular arrangement:
\[
\Phi_B = B \times A
\]
4. **Area of the Loop:** The area, \( A \), of the loop can be calculated as:
\[
A = \pi \times \left(\frac{d}{2}\right)^2
\]
where \( d \) is the diameter of the loop.
This problem applies these principles to find the induced current resulting from the decreasing magnetic field within the solenoid.
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