Step 1: Understanding the Concept:
A current-carrying wire loop placed inside an external magnetic field acts like a magnetic dipole. The magnetic forces acting on opposite segments of the loop create a couple, generating a net rotational torque ($\vec{\tau}$). The magnitude of this torque depends on the orientation of the loop relative to the surrounding magnetic field lines.
Step 2: Key Formula or Approach:
The rotational torque $\vec{\tau}$ exerted on a magnetic dipole with a magnetic moment vector $\vec{M}$ inside a uniform magnetic field $\vec{B}$ is given by the cross product:
$$ \vec{\tau} = \vec{M} \times \vec{B} $$
In scalar terms, the magnitude is:
$$ \tau = MB \sin\theta $$
Where $\theta$ represents the angle between the magnetic moment vector $\vec{M}$ and the magnetic field vector $\vec{B}$. Crucially, the magnetic moment vector $\vec{M}$ is always oriented perpendicular (normal) to the physical plane of the loop.
Step 3: Detailed Explanation:
Let's analyze both statements and their logical connection:
- Evaluating the Reason (R): The fundamental formula for the torque experienced by a magnetic dipole is indeed $\tau = MB \sin\theta$, where $\theta$ is defined as the angle between the area normal vector ($\vec{M}$) and the magnetic field line vectors ($\vec{B}$). Thus, Reason (R) is completely true.
- Evaluating the Assertion (A): The assertion states that the torque reaches its maximum value when the plane of the loop is parallel to the magnetic field lines. Let's calculate the angles for this orientation:
- If the flat plane of the loop sits parallel to $\vec{B}$, its perpendicular normal vector $\vec{M}$ stands at a right angle relative to $\vec{B}$.
- This means $\theta = 90^\circ$.
- Substituting this value into our formula: $\tau = MB \sin(90^\circ) = MB(1) = \tau_{\max}$.
Since $\sin\theta$ reaches its highest value ($1$) at $90^\circ$, the torque is indeed at its absolute maximum under these conditions. Thus, Assertion (A) is completely true.
- Evaluating the Logical Link: Why does the torque reach its maximum value when the loop's plane is parallel to the field? It happens precisely because the torque vector is governed by the sine function of the angle $\theta$ ($\tau = MB \sin\theta$). When the plane is parallel, $\theta$ becomes $90^\circ$, causing $\sin\theta$ to reach its maximum value.
Because the mathematical formula given in the Reason explains the maximum condition stated in the Assertion, the Reason is the correct explanation. This matches option (A).
Step 4: Final Answer:
Both Assertion and Reason are true and Reason is the correct explanation of Assertion.