Question

When a current carrying coil is placed in a uniform magnetic field with its magnetic moment anti-parallel to the field:

• A. Torque on it is maximum,
• B. Torque on it is zero.
• C. Potential energy is maximum.
• D. Dipole is in unstable equilibrium.

Answer 1

Answer: (B), (C), (D)

(a) B, C, D are correct

Answer 2

When a current-carrying coil (magnetic dipole) is placed in a uniform magnetic field, its behavior is described by the torque and potential energy.

Let $\vec{M}$ be the magnetic moment of the coil and $\vec{B}$ be the uniform magnetic field. The angle between $\vec{M}$ and $\vec{B}$ is $\theta$.

1. Torque ($\vec{\tau}$): The torque experienced by the coil is given by the cross product of the magnetic moment and the magnetic field: $\vec{\tau} = \vec{M} \times \vec{B}$ The magnitude of the torque is $\tau = MB \sin\theta$.

2. Potential Energy (U): The potential energy of the coil in the magnetic field is given by the negative dot product of the magnetic moment and the magnetic field: $U = -\vec{M} \cdot \vec{B}$ The potential energy is $U = -MB \cos\theta$.

The problem states that the magnetic moment is anti-parallel to the magnetic field. This means the angle $\theta$ between $\vec{M}$ and $\vec{B}$ is $180^\circ$.

Let's evaluate each statement based on $\theta = 180^\circ$:

• A. Torque on it is maximum. $\tau = MB \sin(180^\circ)$ Since $\sin(180^\circ) = 0$, the torque $\tau = 0$. Maximum torque occurs when $\theta = 90^\circ$ ($\tau_{max} = MB$). Thus, statement A is incorrect.

• B. Torque on it is zero. As calculated above, for $\theta = 180^\circ$, $\tau = 0$. Thus, statement B is correct.

• C. Potential energy is maximum. $U = -MB \cos(180^\circ)$ Since $\cos(180^\circ) = -1$, the potential energy $U = -MB(-1) = +MB$. The potential energy ranges from $-MB$ (minimum, when $\theta = 0^\circ$) to $+MB$ (maximum, when $\theta = 180^\circ$). Thus, statement C is correct.

• D. Dipole is in unstable equilibrium. Equilibrium occurs when the net torque is zero. Since $\tau = 0$ at $\theta = 180^\circ$, the dipole is in equilibrium. To determine the type of equilibrium:

  • Stable equilibrium: Occurs at minimum potential energy ($U = -MB$, when $\theta = 0^\circ$).
  • Unstable equilibrium: Occurs at maximum potential energy ($U = +MB$, when $\theta = 180^\circ$). Since the potential energy is maximum at $\theta = 180^\circ$, the dipole is in unstable equilibrium. Thus, statement D is correct.

Therefore, statements B, C, and D are correct.