Describe torque on a current carrying rectangular loop of wire on a pivot rod when placed in a magnetic field. Give an alternative way of increasing the torque on the coil.

Effect of External Magnetic Field on a Current-Carrying Loop

When a current-carrying loop (or coil) is placed in an external magnetic field, it experiences a torque due to the interaction between the magnetic field and the magnetic moment of the loop. This phenomenon is governed by the principles of electromagnetism, particularly Ampère’s Law and the Lorentz Force.

1. Magnetic Moment of a Current Loop

A current-carrying loop behaves like a magnetic dipole with a magnetic moment (μ) given by: $$\mu =I\cdot A$$ where:

• $I$ = current flowing through the loop (A)
• $A$ = area vector of the loop (m²), perpendicular to the plane of the loop.

The direction of the magnetic moment follows the right-hand rule:

• Curl the fingers of your right hand in the direction of the current.
• The thumb points in the direction of the magnetic moment (μ).

2. Interaction with an External Magnetic Field

When an external magnetic field ($\mathbf{B}$) is applied, the loop experiences a torque (τ) given by: $$\tau =\mu \times B=\mu B\sin \theta$$ where:

• $\theta$ = angle between the magnetic moment (μ) and the external magnetic field (B).
• The torque tends to align the loop’s magnetic moment with the external field (minimum potential energy when $\theta =0$).

3. Direction of Torque

The direction of the torque is determined by the cross product $\mu \times B$:

• If $\theta =0$ (μ parallel to B), no torque ($\tau =0$).
• If $\theta =90^{\circ }$ (μ perpendicular to B), maximum torque ($\tau =\mu B$).
• The torque acts to rotate the loop until μ aligns with B.

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Torque on a Current-Carrying Rectangular Loop in a Magnetic Field

Consider a rectangular loop of wire carrying current $I$, placed in a uniform magnetic field $B$. The loop is pivoted about an axis (e.g., along one of its sides).

Assumptions:

• The loop lies in the xy-plane, with sides of lengths $a$ (along x-axis) and $b$ (along y-axis).
• The magnetic field $B$ is directed at an angle $\theta$ with respect to the normal (z-axis) of the loop.
• The loop is free to rotate about the x-axis (pivot rod).

Step-by-Step Analysis:

1. Magnetic Moment (μ): $$\mu =I\cdot A=I\cdot (a\cdot b)$$ The direction of μ is along the z-axis (perpendicular to the loop).

2. Torque Calculation: The torque on the loop is given by: $$\tau =\mu \times B=\mu B\sin \theta$$

   • If $B$ is parallel to the loop’s plane ($\theta =90^{\circ }$), $\sin \theta =1$, so: $$\tau =IabB$$
   • If $B$ is perpendicular to the loop’s plane ($\theta =0$), $\sin \theta =0$, so $\tau =0$.

3. Force on Individual Sides:

   • The top and bottom sides (length $a$) experience equal and opposite forces $F=IabB$, but since they are parallel, they produce no net torque about the x-axis.
   • The left and right sides (length $b$) experience forces $F=IbB$ in opposite directions, creating a couple that produces torque: $$\tau =F\cdot \text{perpendicular distance}=(IbB)\cdot a=IabB$$ This matches the earlier result.

4. Direction of Rotation:

   • The torque tends to rotate the loop so that its magnetic moment aligns with $B$.
   • If $B$ is into the page, the loop rotates clockwise (as seen from above).

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Alternative Ways to Increase the Torque on the Coil

To maximize the torque ($\tau =\mu B\sin \theta$), we can modify:

1. Increase the Current ($I$):

   • Higher current → larger magnetic moment ($\mu =Iab$) → higher torque.
   • Limitation: Excessive current may cause heating or wire damage.

2. Increase the Area ($A=ab$):

   • Use a larger loop (increase $a$ or $b$) → larger $\mu$ → higher torque.
   • Example: A square loop with $a=b=0.1\text{m}$ has $A=0.01{\text{m}}^2$. Doubling the side length to $0.2\text{m}$ increases $A$ to $0.04{\text{m}}^2$ (4× torque).

3. Increase the Magnetic Field Strength ($B$):

   • Use a stronger magnet or electromagnet → higher $B$ → higher torque.
   • Example: If $B$ is doubled, torque doubles.

4. Optimize the Angle ($\theta$):

   • The torque is maximum when $\theta =90^{\circ }$ (μ perpendicular to $B$).
   • If the loop is initially at $\theta =30^{\circ }$, rotating it to $90^{\circ }$ increases torque by $\sin 90^{\circ }/\sin 30^{\circ }=2$.

5. Use Multiple Turns (Helical Coil):

   • A coil with $N$ turns has a magnetic moment $N$ times larger: $${\mu }_{\text{total}}=N\cdot I\cdot A$$
   • Torque becomes: $$\tau =NIAB\sin \theta$$
   • Example: A 10-turn coil has 10× the torque of a single loop.

6. Shape Optimization:

   • A circular loop has a higher torque-to-area ratio than a rectangular loop for the same perimeter.
   • Example: For a given wire length, a circular loop maximizes $A$, hence $\mu$.

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Practical Example (Numerical Problem)

Problem: A rectangular loop of dimensions $0.2\text{m}\times 0.1\text{m}$ carries a current of $5\text{A}$. It is placed in a magnetic field of $0.5\text{T}$ at an angle of $60^{\circ }$ to the loop’s normal. Calculate the torque.

Solution:

1. Calculate the area ($A$): $$A=0.2\times 0.1=0.02{\text{m}}^2$$
2. Calculate the magnetic moment ($\mu$): $$\mu =I\cdot A=5\times 0.02=0.1{\text{A⋅m}}^2$$
3. Calculate the torque ($\tau$): $$\tau =\mu B\sin \theta =0.1\times 0.5\times \sin 60^{\circ }$$ $$\sin 60^{\circ }=\frac{\sqrt{3}}{2}\approx 0.866$$ $$\tau =0.1\times 0.5\times 0.866=0.0433\text{N⋅m}$$ Final Answer: $\tau =0.043\text{N⋅m}$

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Key Takeaways

• A current loop in a magnetic field experiences a torque that tends to align its magnetic moment with the field.
• The torque depends on current, area, magnetic field strength, and angle.
• Torque can be increased by:
  • Increasing current ($I$).
  • Increasing loop area ($A$).
  • Using a stronger magnetic field ($B$).
  • Orienting the loop perpendicular to $B$ ($\theta =90^{\circ }$).
  • Using multiple turns ($N$).
  • Optimizing the loop shape (e.g., circular).