A flat, 192-turn, current-carrying loop is immersed in a uniform magnetic field. The area of the loop is 8.37 cm ^2 and the angle between its magnetic dipole moment and the field is 36.5 ∘ . Find the strength B of the magnetic field that causes a torque of 2.89×10 ^−5 N⋅m to act on the loop when a current of 3.87 mA flows in it.
The Correct Answer and Explanation is :
To solve for the magnetic field strength ( B ) that causes a torque ( \tau ) on a current-carrying loop, we can use the formula for torque in a magnetic field:
[
\tau = n \cdot I \cdot A \cdot B \cdot \sin(\theta)
]
where:
- ( \tau ) = torque (N·m)
- ( n ) = number of turns in the loop
- ( I ) = current (A)
- ( A ) = area of the loop (m²)
- ( B ) = magnetic field strength (T)
- ( \theta ) = angle between the magnetic dipole moment and the magnetic field (radians)
Given Data:
- Number of turns, ( n = 192 )
- Current, ( I = 3.87 \, \text{mA} = 3.87 \times 10^{-3} \, \text{A} )
- Area, ( A = 8.37 \, \text{cm}^2 = 8.37 \times 10^{-4} \, \text{m}^2 )
- Torque, ( \tau = 2.89 \times 10^{-5} \, \text{N} \cdot \text{m} )
- Angle, ( \theta = 36.5^\circ )
Converting the Angle:
Convert the angle from degrees to radians for calculations:
[
\theta = 36.5^\circ \times \frac{\pi}{180} \approx 0.637 \, \text{radians}
]
Plugging in Values:
Now we can rearrange the formula to solve for ( B ):
[
B = \frac{\tau}{n \cdot I \cdot A \cdot \sin(\theta)}
]
Calculate ( B ):
Calculating ( \sin(0.637) ):
[
\sin(0.637) \approx 0.6018
]
Now substituting all values:
[
B = \frac{2.89 \times 10^{-5}}{192 \cdot (3.87 \times 10^{-3}) \cdot (8.37 \times 10^{-4}) \cdot (0.6018)}
]
Calculating the denominator:
[
= 192 \cdot 3.87 \times 10^{-3} \cdot 8.37 \times 10^{-4} \cdot 0.6018 \approx 1.150 \times 10^{-7}
]
Now substitute back to find ( B ):
[
B = \frac{2.89 \times 10^{-5}}{1.150 \times 10^{-7}} \approx 251.3 \, \text{T}
]
Final Answer:
Thus, the magnetic field strength ( B ) is approximately 251.3 T.
Explanation:
In this problem, we utilized the relationship between torque, magnetic field, and the parameters of a current-carrying loop. The torque on a loop in a magnetic field arises from the interaction of the current flowing through the loop and the magnetic field itself. The area of the loop, the number of turns, the current, and the angle between the magnetic moment and the magnetic field all contribute to the overall torque experienced by the loop. This relationship is critical in many applications, including electric motors, generators, and magnetic sensors, where understanding and manipulating the torque is essential for efficient design and operation.