A 50-turn rectangular coil of wire of dimensions 10.0 cm by 20.0 cm lies in a horizontal plane

step 1 Here is given problem a rectangular current carrying coil will be under the influence of two tor

A 50-turn rectangular coil of wire of dimensions 10.0 cm by...

A 50-turn rectangular coil of wire of dimensions $10.0 \mathrm{~cm}$ by $20.0 \mathrm{~cm}$ lies in a horizontal plane, as shown in the figure. The axis of rotation of the coil is aligned north and south. It carries a current $i=1.00 \mathrm{~A}$, and is in a magnetic field pointing from west to east. A mass of $50.0 \mathrm{~g}$ hangs from one side of the loop. Determine the strength the magnetic field has to have to keep the loop in the horizontal orientation.

Key Concepts

Magnetic Dipole Moment

The magnetic dipole moment is a measure of the strength and orientation of a current-carrying loop's magnetic effects. It is defined as the product of the current, the number of turns, and the area of the loop. This quantity is fundamental in analyzing how a coil interacts with an external magnetic field and determines the magnitude of the torque exerted on the coil.

Magnetic Torque

Magnetic torque is the twisting force experienced by a magnetic dipole when placed in an external magnetic field. It is given by the cross product of the dipole moment and the magnetic field vector, with a magnitude equal to the product of the magnetic dipole moment, the magnetic field strength, and the sine of the angle between them. This relationship explains how the magnetic field tends to align the dipole with its own direction.

Torque Equilibrium

Torque equilibrium occurs when the net torque acting on a system is zero, meaning there is no angular acceleration. In problems involving current loops and external forces, achieving torque equilibrium means that the magnetic torque is balanced by any other torques present, such as gravitational torque. This condition is often used to calculate unknown quantities such as the magnetic field strength required to maintain a specific orientation.

Gravitational Torque

Gravitational torque arises from the gravitational force acting at a distance (lever arm) from the pivot point of a system. In systems where a mass is attached to a current loop, this torque tends to rotate the system due to the weight of the mass acting at a distance from the axis of rotation. Understanding gravitational torque is essential when it is balanced against other torques, such as magnetic torque, to determine equilibrium conditions.

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