Why is torque applied on a current-carrying coil placed in a magnetic field? Discuss in detail. What are the factors on which torque depends?

When a current-carrying coil is placed in a magnetic field, the interaction between the magnetic field and the current in the coil generates a torque. This torque tends to rotate the coil, and it’s the fundamental working principle behind electric motors.

Why is Torque Applied on a Current-Carrying Coil?

A current-carrying coil in a magnetic field experiences forces due to the interaction of the magnetic field with the moving charges (electrons) in the conductor. Let’s break this down:

1. Magnetic Force on the Coil’s Sides: Each segment of the coil carrying current experiences a force because a current-carrying conductor in a magnetic field experiences a force (as per the formula F=BILsin⁡θ

2. Direction of Forces on Different Sides of the Coil: In a rectangular coil, the two opposite sides (parallel to the magnetic field) experience forces in opposite directions, while the other two sides (perpendicular to the magnetic field) experience forces that also act in opposite directions. These opposing forces create a torque about the center of the coil, causing it to rotate.

3. Torque Generation: The net force on the coil does not cancel out because the forces on opposite sides of the coil are not in the same line. The forces on these sides create a couple, which generates a torque. This torque tends to rotate the coil about its axis, which is perpendicular to the plane of the coil.

Formula for Torque:

The torque τ experienced by a current-carrying coil in a magnetic field is given by:

τ=BIAsin⁡θ

Where:

• τ = Torque on the coil (in newton-meters, Nm)
• $B$ = Magnetic field strength (in teslas, T)
• $I$ = Current in the coil (in amperes, A)
• $A$ = Area of the coil (in square meters, m²)
• θ= Angle between the normal to the plane of the coil and the magnetic field

Alternatively, if the coil has $N$ turns, the formula becomes:

τ=NBIAsin⁡θ

Where:

• $N$ = Number of turns in the coil

Factors on Which Torque Depends:

The torque on a current-carrying coil depends on the following factors:

1. Magnetic Field Strength (B): A stronger magnetic field increases the torque. This is because the force acting on the current-carrying conductor increases with the strength of the magnetic field.

2. Current (I): A larger current flowing through the coil increases the force on each segment of the coil, leading to a greater torque.

3. Area of the Coil (A): The larger the area of the coil, the greater the torque. The area $A$ is proportional to the distance between opposite sides of the coil, and the force acting on a larger area will generate a larger torque.

4. Angle (θ): The torque is maximized when the plane of the coil is perpendicular to the magnetic field (i.e., when θ=90∘). If the plane of the coil is parallel to the magnetic field, the torque will be zero (i.e., when θ=0∘).

5. Number of Turns (N): For a coil with multiple turns, the torque increases with the number of turns $N$ because each turn experiences a similar torque. More turns mean more effective interaction between the coil and the magnetic field.