Q.Arrive at an expression for torque on a rectangular current loop in a uniform magnetic field.

The torque on a magnetic dipole is $\vec{\tau} = \vec{m}\times\vec{B}$, analogous to torque on an electric dipole $\vec{p}\times\vec{E}$.

When a magnetic dipole of moment $\vec{m}$ is placed in a uniform magnetic field $\vec{B}$, it experiences a torque that tends to align it with the field. This torque is given by the vector product:

$\vec{\tau} = \vec{m}\times\vec{B}$

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Magnetic moment M = I·(πr²), so I = M/(πr²) ∝ M/r². Substituting the given ratios gives I₁ : I₂ = 2 : 1. Option (b).

Step 1 — magnetic moment of a current loop (NCERT/CBSE Class 12 Physics, Moving Charges and Magnetism): M = I·A = I·πr².

Step 2 — so current I = M/(πr²), i.e. I ∝ M/r².

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Magnetic moment m = IA = Iπr²; equal m with r ratio 1:2 forces the current ratio to be 4:1 (inverse of the area ratio).

Magnetic moment of a current loop is $m = I\pi r^2$. Let the radii be $r_1:r_2 = 1:2$ and the moments be equal, $m_1=m_2$: …

increasing (the torque is directly proportional to the area of the loop). …

A current loop behaves like a magnetic dipole; the torque on it is the cross product of its magnetic moment and the field, τ = m × B.

A planar current loop carrying current I and enclosing area A has a magnetic (dipole) moment $\vec{m} = I\vec{A}$, directed along the normal to the loop (right-hand rule).

When this dipole is placed in a uniform magnetic field $\vec{B}$, the two sides of the loop carry equal and opposite forces that form a couple. The resulting torque is

$\vec{\tau} = \vec{m} \times \vec{B}, \qquad \tau = mB\sin\theta$

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Torque on a magnetic dipole (current loop) in a magnetic field.

A current-carrying coil of magnetic dipole moment $\vec{m}$ placed in a uniform magnetic field $\vec{B}$ experiences a torque that tends to align $\vec{m}$ with $\vec{B}$:

$\vec{\tau} = \vec{m}\times\vec{B}$ …

For a fixed wire length $L$, a circle encloses the maximum area, so the square loop has less area than the circular loop. Since torque $\tau = NIAB\sin\theta$ depends directly on area, the square loop experiences less torque — making Assertion false and Reason false as well. The correct choice is (D).

Torque on a current loop: τ⃗ = M⃗ × B⃗.

A current loop of magnetic moment M⃗ = NIA n̂ placed in a uniform magnetic field B⃗ experiences a torque

$\vec{\tau} = \vec{M}\times\vec{B},$

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The net magnetic force on any closed current loop placed in a uniform magnetic field is always zero.

For a current loop in a UNIFORM magnetic field B, the net force is F = I∮dl×B = I(∮dl)×B. Since the vector sum of all the line elements around a closed loop is zero (∮dl = 0), the net force is always zero — regardless of the loop's shape or its orientation relative to B. (A net torque can still act if …