Question

A straight wire loop carrying a current \( I \) sits flat inside a uniform, external magnetic field \( \vec{B} \). Under what orientation angle \( \theta \) between the normal vector of the loop surface and the magnetic field lines will the loop experience maximum torque?

• A. \( \theta = 0^\circ \)
• B. \( \theta = 45^\circ \)
• C. \( \theta = 90^\circ \)
• D. \( \theta = 180^\circ \)

Hint:

Be sure to read carefully whether the angle given in the question is measured from the normal to the loop face (\(\theta\)) or from the flat plane of the coil itself (\(\alpha\)). Note that \(\theta + \alpha = 90^\circ\), so torque is maximized when the plane of the coil is parallel to the field lines (\(\alpha = 0^\circ\), \(\theta = 90^\circ\)).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
A current-carrying loop in a magnetic field experiences magnetic forces on its segments.
If the loop is placed such that the forces on opposite sides act in opposite directions but not along the same line, a "couple" is formed, which generates a turning effect or torque (\(\tau\)).
The amount of torque depends on how the loop's face is oriented relative to the field lines.
Step 2: Key Formula or Approach:
The magnitude of the torque is given by the formula:
\[ \tau = MB \sin \theta \]
Where:
\(M = NIA\) is the magnetic dipole moment.
\(B\) is the external magnetic field strength.
\(\theta\) is the angle between the normal vector of the loop and the magnetic field lines.
Step 3: Detailed Explanation:
The torque \(\tau\) is a function of the sine of the angle \(\theta\).
To find the maximum possible torque, we must maximize the value of \(\sin \theta\).
The sine function reaches its maximum value of \(1\) when the angle is \(90^{\circ}\):
\[ \sin(90^{\circ}) = 1 \]
At \(\theta = 90^{\circ}\), the torque becomes \(\tau = MB\).
Physically, an angle of \(\theta = 90^{\circ}\) means the normal vector (perpendicular to the loop face) is perpendicular to the field.
This happens when the actual flat plane of the loop is parallel to the magnetic field lines. In this orientation, the forces are most effective at causing rotation.
Conversely, if \(\theta = 0^{\circ}\) (plane is perpendicular to field), \(\sin(0^{\circ}) = 0\) and the torque is zero.
Step 4: Final Answer:
The loop experiences maximum torque when the angle \(\theta\) is \(90^{\circ}\).