Question

A plane wire loop of irregular shape is situated so that part of it is in a uniform magnetic field B (in Fig. 5.57 the field occupies the shaded region, and points perpendicular to the plane of the loop). The loop carries a current I. Show that the net magnetic force on the loop is$\mathbf{F}\mathbf{=}\mathbf{IB\omega }$, where$\mathit{\omega }$is the chord subtended. Generalize this result to the case where the magnetic field region itself has an irregular shape. What is the direction of the force?

Short answer

The net magnetic force on the loop is$F=IB\omega$ .

The force acts in the direction perpendicular to the chord subtended by the loop.

Step-by-step solution

Identification of the given data

The given data is listed below as:

• The uniform magnetic field of the wire loop is,$B$
• The current carried by the loop is,$I$
• The chord subtended by the loop is,$\omega$

Significance of magnetic field

The magnetic field is described as the region around a particular magnet. The magnetic field is also beneficial for understanding distribution of magnetic force around a particular magnetic material.

Determination of the net magnetic force on the loop

The equation of the net magnetic force on the loop is expressed as:

$F=I\int \left(dI\times B\right)$

Here,$I$is current,$dI$is the increase in the length and$B$is the uniform magnetic field.

As the uniform magnetic field is constant, then the equation of the net magnetic force can be expressed as:

$F=IB\int \left(dI\right)$ …(i)

The entering and the leaving point of the wire inside the magnetic field is same. So, the chord subtended by the loop and the uniform magnetic field are perpendicular to each other. Hence, the equation of the integration of the increase in the length can be expressed as:

$\int \left(dI\right)=\omega$

Here,$\omega$ is the chord subtended by the loop.

Substitute the above equation in equation (i).

$F=IB\omega$

Thus, the net magnetic force on the loop is $F=IB\omega$.

The force acts in the direction perpendicular to the chord subtended by the loop.