Torque and Magnetic Moment on a Rectangular Loop in a Uniform Magnetic Field

Описание: RECTANGULAR LOOP IN UNIFORM FIELD , FORCE AND TORQUE ON THE CLOSED LOOP , TORQUE ON THE CLOSED LOOP , MAGNETIC MOMENT AND TORQUE


RECTANGULAR LOOP IN UNIFORM FIELD


A rectangular loop carrying current I is in a uniform magnetic field parallel to the plane of the loop, no force is exerted on the arms AD and BC of the rectangle.

But arms AB and CD experience forces of equal magnitude F but in the opposite directions perpendicular to the plane.


FORCE AND TORQUE ON THE CLOSED LOOP


The forces on the arms AB and CD of the loop sum up to zero, but contribute to the torque on the loop about its rotation axis.

Magnitude of the force on these arms is

F = IbB (B: magnetic field);

\Gamma (magnitude of the torque about the axis)

= \frac{Fa}{2} + \frac{Fa}{2} = Fa = IabB = IAB

(area of the loop = A = ab).

Here, b & a are length and width of the loop


TORQUE ON THE CLOSED LOOP


Now considering, the loop rotates making an angle \theta between the field and normal vector to the coil (i.e., Area vector \vec{A}), the forces on arms AC and BD don't contribute to the torque.

Net torque is due to the forces on arms AB and CD.

\Gamma = F(\frac{a}{2})\sin\theta + F(\frac{a}{2})\sin\theta

\Rightarrow IabB\sin\theta = IAB\sin\theta \Rightarrow \vec{\Gamma} = I(\vec{A} \times \vec{B})


MAGNETIC MOMENT AND TORQUE


As we know,

\vec{\Gamma} = I(\vec{A} \times \vec{B})

Torque \vec{\Gamma} can be defined as cross product of \vec{m} and \vec{B}.

\vec{m} is magnetic moment vector, defined as \vec{m} = I\vec{A} (where \vec{A} is area vector)

Hence, torque on a current loop can be expressed as \vec{\Gamma} = I(\vec{A} \times \vec{B}) = \vec{m} \times \vec{B}.