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

Concept: When a current-carrying loop is placed inside a uniform magnetic field, the magnetic forces acting on opposite sides of the loop create a couple, generating a net turning effect or torque (\( \tau \)). The magnitude of this torque depends on the orientation of the loop relative to the field lines: \[ \tau = M B \sin\theta \] where \( M = NIA \) is the magnetic dipole moment of the loop, \( B \) is the magnetic field strength, and \( \theta \) is the angle between the normal vector to the plane of the loop and the magnetic field lines.

Step 1: Maximize the sine function term. In the torque equation, the variables \( M \) and \( B \) are fixed scale values. The torque reaches its maximum value when the angular term \( \sin\theta \) reaches its absolute maximum possible value: \[ \sin\theta = 1 \] The angle that satisfies this condition within standard orientation geometry is: \[ \theta = 90^\circ \]

Step 2: Interpret the physical orientation of the loop face. An angle of \( \theta = 90^\circ \) means the normal vector of the loop is perpendicular to the field lines. This occurs when the flat plane of the coil loop runs parallel to the surrounding magnetic field lines. In this orientation, the moment arm for the magnetic forces is at its longest, producing the maximum possible torque (\( \tau_{\text{max}} = NIAB \)). Conversely, when \( \theta = 0^\circ \), the loop is perpendicular to the field lines, resulting in zero torque (\(\sin(0^\circ) = 0\)).