A magnetic field exerts a maximum torque to on a flat, current-carrying rectangular coil of N loops. What is the maximum torque the magnetic field exerts on a coil with 3 times the current, 0.8 times the area, and 4 times the number of loops in the coil? _) TO Tnew
A magnetic field exerts a maximum torque to on a flat, current-carrying rectangular coil of N loops. What is the maximum torque the magnetic field exerts on a coil with 3 times the current, 0.8 times the area, and 4 times the number of loops in the coil? _) TO Tnew
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![**Problem Statement:**
A magnetic field exerts a maximum torque \( \tau_0 \) on a flat, current-carrying rectangular coil of \( N \) loops. What is the maximum torque the magnetic field exerts on a coil with 3 times the current, 0.8 times the area, and 4 times the number of loops in the coil?
\[ \tau_{\text{new}} = (\ \ \ \ \ ) \tau_0 \]
**Solution Explanation:**
To find the new maximum torque (\( \tau_{\text{new}} \)), we use the formula for torque in a magnetic field:
\[ \tau = n \cdot I \cdot A \cdot B \cdot \sin(\theta) \]
Where:
- \( n \) is the number of loops,
- \( I \) is the current,
- \( A \) is the area of the coil,
- \( B \) is the magnetic field strength,
- \( \theta \) is the angle between the magnetic field and the normal to the coil (assumed to be 90 degrees for maximum torque, hence \(\sin(\theta) = 1\)).
Given:
- The initial torque is \( \tau_0 \).
- For the new coil, the current is increased by a factor of 3, so \( I_{\text{new}} = 3I \).
- The area is reduced to 0.8 times, so \( A_{\text{new}} = 0.8A \).
- The number of loops is increased by a factor of 4, so \( n_{\text{new}} = 4n \).
Substituting these values into the torque formula for the new coil:
\[ \tau_{\text{new}} = n_{\text{new}} \cdot I_{\text{new}} \cdot A_{\text{new}} \cdot B \]
\[ \tau_{\text{new}} = (4n) \cdot (3I) \cdot (0.8A) \cdot B \]
\[ \tau_{\text{new}} = 4 \cdot 3 \cdot 0.8 \cdot n \cdot I \cdot A \cdot B \]
\[ \tau_{\text{new}} = 9.6 \cdot (n \cdot I \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F71da004f-23dc-47ac-9c54-84cf84b3724a%2F178b0d16-0445-4974-b0f5-a7c863b30c94%2F74fbync_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
A magnetic field exerts a maximum torque \( \tau_0 \) on a flat, current-carrying rectangular coil of \( N \) loops. What is the maximum torque the magnetic field exerts on a coil with 3 times the current, 0.8 times the area, and 4 times the number of loops in the coil?
\[ \tau_{\text{new}} = (\ \ \ \ \ ) \tau_0 \]
**Solution Explanation:**
To find the new maximum torque (\( \tau_{\text{new}} \)), we use the formula for torque in a magnetic field:
\[ \tau = n \cdot I \cdot A \cdot B \cdot \sin(\theta) \]
Where:
- \( n \) is the number of loops,
- \( I \) is the current,
- \( A \) is the area of the coil,
- \( B \) is the magnetic field strength,
- \( \theta \) is the angle between the magnetic field and the normal to the coil (assumed to be 90 degrees for maximum torque, hence \(\sin(\theta) = 1\)).
Given:
- The initial torque is \( \tau_0 \).
- For the new coil, the current is increased by a factor of 3, so \( I_{\text{new}} = 3I \).
- The area is reduced to 0.8 times, so \( A_{\text{new}} = 0.8A \).
- The number of loops is increased by a factor of 4, so \( n_{\text{new}} = 4n \).
Substituting these values into the torque formula for the new coil:
\[ \tau_{\text{new}} = n_{\text{new}} \cdot I_{\text{new}} \cdot A_{\text{new}} \cdot B \]
\[ \tau_{\text{new}} = (4n) \cdot (3I) \cdot (0.8A) \cdot B \]
\[ \tau_{\text{new}} = 4 \cdot 3 \cdot 0.8 \cdot n \cdot I \cdot A \cdot B \]
\[ \tau_{\text{new}} = 9.6 \cdot (n \cdot I \
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