Problem

A $0.2 \, \mathrm{m}$ long wire is bent into a circular shape and placed in a uniform magnetic field of $2 \, \mathrm{T}$. If the current in the wire is $20 \, \mathrm{mA}$, find the maximum torque acting on the loop.

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Data

• Length of the wire: $L = 0.2 \, \mathrm{m}$,
• Magnetic field: $B = 2 \, \mathrm{T}$,
• Current: $I = 20 \, \mathrm{mA} = 20 \times 10^{-3} \, \mathrm{A}$.

To find:

• Maximum torque: $\tau_{\max} = ?$.

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Prerequisite Concepts

1. Circumference of a Circle: If the wire is bent into a circular shape, the circumference is equal to the wire’s length:

$$L = 2 \pi r$$

Rearrange for the radius:

$$r = \frac{L}{2 \pi}$$

2. Area of a Circle: The area of a circle is given by:

$$A = \pi r^2$$

3. Torque on a Current-Carrying Loop: The maximum torque acting on a current-carrying loop in a magnetic field is given by:

$$\tau_{\max} = I B A$$

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Solution

1. Calculate the Radius: Using:

$$r = \frac{L}{2 \pi}$$

Substituting the values:

$$r = \frac{0.2}{2 \times 3.14} = 0.0318 \, \mathrm{m}.$$

2. Calculate the Area: Using:

$$A = \pi r^2$$

Substituting the values:

$$A = 3.14 \times (0.0318)^2 = 3.18 \times 10^{-3} \, \mathrm{m}^2.$$

3. Calculate the Maximum Torque: Using:

$$\tau_{\max} = I B A$$

Substituting the values:

$$\tau_{\max} = (20 \times 10^{-3}) \cdot 2 \cdot 3.18 \times 10^{-3}$$

Simplify:

$$\tau_{\max} = 1.272 \times 10^{-4} \, \mathrm{Nm}.$$

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Answer

The maximum torque acting on the loop is:

$$\tau_{\max} = 1.272 \times 10^{-4} \, \mathrm{Nm}.$$