A rectangular loop of wire with sides H=26cm and W=65cm is located in a region containing a constant magnetic field B=1.12T that is aligned with the positive y-axis. The loop carries a current I=205mA . The plane of the loop is inclined at an angle θ=28
∘
with respect to the x-axis.
- What is μ
x
, the x-component of the magnetic moment vector of the loop? 2. What is μ
y
, the y-component of the magnetic moment vector of the loop? 3. What is τ
z
, the z-component of the torque exerted on the loop? 4. What is F
bc
, the magnitude of the force exerted on segment bc of the loop? 5. What is the direction of the force that is exerted on segment bc of the loop? – A. Along the negative x-direction – B. Along the negative y-direction – C. Along the positive x-direction – D. Along the positive y-direction – E. None of the above
The Correct Answer and Explanation is :
To analyze the rectangular loop of wire carrying a current in a magnetic field, we can derive the necessary quantities step by step.
1. Magnetic Moment ( \mu_x )
The magnetic moment ( \mu ) of a rectangular loop is given by the formula:
[
\mu = I \cdot A
]
where ( I ) is the current and ( A ) is the area of the loop. The area ( A ) can be calculated as:
[
A = H \times W = 0.26 \, \text{m} \times 0.65 \, \text{m} = 0.169 \, \text{m}^2
]
Thus,
[
\mu = 0.205 \, \text{A} \times 0.169 \, \text{m}^2 = 0.034645 \, \text{A} \cdot \text{m}^2
]
Since the loop is inclined at an angle ( \theta ) with respect to the x-axis, we can find the components:
[
\mu_x = \mu \cdot \cos(\theta) = 0.034645 \cdot \cos(28^\circ) \approx 0.0306 \, \text{A} \cdot \text{m}^2
]
2. Magnetic Moment ( \mu_y )
[
\mu_y = \mu \cdot \sin(\theta) = 0.034645 \cdot \sin(28^\circ) \approx 0.0164 \, \text{A} \cdot \text{m}^2
]
3. Torque ( \tau_z )
The torque ( \tau ) exerted on a magnetic moment in a magnetic field is given by:
[
\tau = \mu \times B
]
The magnitude of the torque about the z-axis can be calculated using:
[
\tau_z = \mu \cdot B \cdot \sin(\theta) = 0.034645 \cdot 1.12 \cdot \sin(28^\circ) \approx 0.01065 \, \text{N} \cdot \text{m}
]
4. Force on Segment ( bc ) (( F_{bc} ))
The force on a current-carrying segment in a magnetic field is given by:
[
F = I \cdot L \times B
]
Where ( L ) is the length of the segment and points in the direction of the current. The length of segment ( bc ) is ( H = 0.26 \, \text{m} ), and since it is aligned along the width in the x-direction, the force is:
[
F_{bc} = I \cdot H \cdot B = 0.205 \cdot 0.26 \cdot 1.12 \approx 0.059 \, \text{N}
]
5. Direction of the Force
Using the right-hand rule, since the current ( I ) flows upward in the y-direction (the direction of the magnetic field) and the magnetic field is along the positive y-axis, the force on segment ( bc ) will be directed along the negative x-direction. Thus, the correct answer is:
- A. Along the negative x-direction
Conclusion
This analysis of the magnetic properties of a rectangular loop in a magnetic field highlights the importance of understanding the relationships between current, magnetic fields, and forces acting on current-carrying conductors. Each step provides insight into how the geometry and orientation of the loop impact the magnetic moment and resulting forces, making this a fundamental topic in electromagnetism.