Let's analyze the problem of a square current loop placed in a magnetic field and calculate the torque it experiences. This involves understanding the interaction between a current-carrying loop and a magnetic field, a fundamental concept in electromagnetism.

Understanding Torque on a Current Loop in a Magnetic Field

A current-carrying loop in a magnetic field behaves like a magnetic dipole. It experiences a torque that tends to align its magnetic dipole moment with the external magnetic field. The torque experienced by a current loop in a uniform magnetic field is given by the formula:

\(\vec{\tau} = \vec{\mu} \times \vec{B}\)

where:

• \(\vec{\tau}\) is the torque vector.
• \(\vec{\mu}\) is the magnetic dipole moment vector of the loop.
• \(\vec{B}\) is the magnetic field vector.

The magnitude of the magnetic dipole moment (\(\mu\)) for a planar loop is given by:

\(\mu = IA\)

where:

• \(I\) is the current flowing through the loop.
• \(A\) is the area of the loop.

The direction of the magnetic dipole moment vector \(\vec{\mu}\) is perpendicular to the plane of the loop, given by the right-hand rule (if the fingers curl in the direction of the current, the thumb points in the direction of \(\vec{\mu}\)).

The magnitude of the torque is given by:

\(\tau = \mu B \sin\theta\)

where \(\theta\) is the angle between the magnetic dipole moment vector \(\vec{\mu}\) and the magnetic field vector \(\vec{B}\).

Calculating Torque for the Square Loop

We are given the following information:

• Shape of the loop: Square
• Side length (\(L\)): 1 cm = 0.01 m
• Current (\(I\)): 10 A
• Magnetic field strength (\(B\)): 0.2 T
• Direction of magnetic field: Parallel to the plane of the loop.

Step-by-Step Calculation:

1. Calculate the Area of the Loop: The loop is a square with side length \(L = 0.01\) m. The area \(A\) is: \[A = L^2 = (0.01 \text{ m})^2 = 0.0001 \text{ m}^2 = 1 \times 10^{-4} \text{ m}^2\]
2. Calculate the Magnetic Dipole Moment: Using the formula \(\mu = IA\): \[\mu = (10 \text{ A})(1 \times 10^{-4} \text{ m}^2) = 10 \times 10^{-4} \text{ A m}^2 = 1 \times 10^{-3} \text{ A m}^2\]
3. Determine the Angle between \(\vec{\mu}\) and \(\vec{B}\): The magnetic dipole moment vector \(\vec{\mu}\) is always perpendicular to the plane of the loop. The problem states that the magnetic field \(\vec{B}\) is parallel to the plane of the loop. Therefore, the angle \(\theta\) between \(\vec{\mu}\) and \(\vec{B}\) is \(90^\circ\). \[\theta = 90^\circ\] \[\sin\theta = \sin(90^\circ) = 1\]
4. Calculate the Magnitude of the Torque: Using the formula \(\tau = \mu B \sin\theta\): \[\tau = (1 \times 10^{-3} \text{ A m}^2)(0.2 \text{ T})(1)\] \[\tau = 0.2 \times 10^{-3} \text{ Nm}\] \[\tau = 2 \times 10^{-4} \text{ Nm}\]

Result of the Torque Calculation

The torque experienced by the square loop is \(2 \times 10^{-4}\) Nm.

Let's compare this result with the given options:

Our calculated value, \(2 \times 10^{-4}\) Nm, matches Option 2.

Revision Table: Torque on a Current Loop

Additional Information: Torque and Magnetic Potential Energy

Besides experiencing a torque, a current loop in a magnetic field also possesses magnetic potential energy. The potential energy (\(U\)) of a magnetic dipole in a magnetic field is given by:

\(U = -\vec{\mu} \cdot \vec{B} = -\mu B \cos\theta\)

The torque tends to rotate the loop towards a position of minimum potential energy. Minimum potential energy occurs when \(U\) is minimum, which happens when \(\cos\theta\) is maximum, i.e., \(\theta = 0^\circ\). This is when the magnetic dipole moment \(\vec{\mu}\) is aligned parallel to the magnetic field \(\vec{B}\).

The maximum torque occurs when \(\sin\theta\) is maximum, i.e., \(\theta = 90^\circ\). This is the case in this problem, where the magnetic field is parallel to the plane of the loop, making \(\vec{\mu}\) perpendicular to \(\vec{B}\).

When the loop is aligned with the field (\(\theta = 0^\circ\) or \(180^\circ\)), the torque is zero, but the potential energy is at its minimum or maximum respectively. When \(\vec{\mu}\) is parallel to \(\vec{B}\) (\(\theta = 0^\circ\)), it is a stable equilibrium position. When \(\vec{\mu}\) is antiparallel to \(\vec{B}\) (\(\theta = 180^\circ\)), it is an unstable equilibrium position.