A flat 145-turn current-carrying loop is immersed in a uniform magnetic field. The area of the loop is 6.75 cm2 and the angle between its magnetic dipole moment and the field is 31.5°. Find the strength of the magnetic field that causes a torque of 1.33 × 10-5 N·m to act on the loop when a current of 2.11 mA flows in it.
The correct Answer and Explanation is:
To find the strength of the magnetic field ( B ) that causes a torque of ( 1.33 \times 10^{-5} \, \text{N} \cdot \text{m} ) to act on a current-carrying loop, we can use the following formula for the torque ( \tau ) on a current-carrying loop in a magnetic field:
[
\tau = \mu \cdot B \cdot \sin(\theta)
]
where:
- ( \tau ) is the torque,
- ( \mu ) is the magnetic dipole moment of the loop,
- ( B ) is the magnetic field strength,
- ( \theta ) is the angle between the magnetic dipole moment and the magnetic field.
Step 1: Calculate the Magnetic Dipole Moment ( \mu )
The magnetic dipole moment ( \mu ) of a current-carrying loop is given by:
[
\mu = I \cdot A \cdot N
]
where:
- ( I ) is the current (in amperes),
- ( A ) is the area of the loop (in square meters),
- ( N ) is the number of turns of the loop.
Given:
- ( I = 2.11 \, \text{mA} = 2.11 \times 10^{-3} \, \text{A} ),
- ( A = 6.75 \, \text{cm}^2 = 6.75 \times 10^{-4} \, \text{m}^2 ),
- ( N = 145 ) turns.
Now, calculate the magnetic dipole moment:
[
\mu = (2.11 \times 10^{-3}) \cdot (6.75 \times 10^{-4}) \cdot 145 = 2.14 \times 10^{-4} \, \text{A} \cdot \text{m}^2
]
Step 2: Solve for the Magnetic Field ( B )
Using the torque formula:
[
\tau = \mu \cdot B \cdot \sin(\theta)
]
Rearranging to solve for ( B ):
[
B = \frac{\tau}{\mu \cdot \sin(\theta)}
]
Substitute the known values:
[
B = \frac{1.33 \times 10^{-5}}{(2.14 \times 10^{-4}) \cdot \sin(31.5^\circ)}
]
First, calculate ( \sin(31.5^\circ) \approx 0.523 ), and then:
[
B = \frac{1.33 \times 10^{-5}}{(2.14 \times 10^{-4}) \cdot 0.523} \approx 1.17 \, \text{T}
]
Final Answer:
The strength of the magnetic field is approximately 1.17 T.