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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.
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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