91影视

• Figure of three circuits is given.
• Same current is flowing through each circuit.

Formulae are as follows:

Find the net magnetic field at the center of each circuit using the formula for the magnetic field at the center of a circular loop. Comparing them rank the givencircuits according to the magnitude of the magnetic field at the center.

Where, B is a magnetic field, R is the radius, iis current,饾泹is permeability.

The magnitude of magnetic field at the center of arc is given by,

$B=\frac{{渭}_{0}i}{4蟺R}$

Let the magnetic field at the center of arc due to arc of radius r be B₁ and due to arc of radius 2r be B₂,and due to arc of radius 3r be B₃ .

Then,

The net magnetic field at the center of circuit (a) is,

$B=B_1+B_3=\frac{{渭}_{0}i蟺}{4蟺r}+\frac{{渭}_{0}i蟺}{12蟺r}=\frac{{渭}_{0}i}{3r}$

The net magnetic field at the center of circuit (b) is,

$B=B_1-B_3=\frac{{渭}_{0}i蟺}{4蟺r}-\frac{{渭}_{0}i蟺}{12蟺r}=\frac{{渭}_{0}i}{6r}$

The net magnetic field at the center of circuit (c) is,

$B=B_1+B_2+B_3=\frac{{渭}_{0}i蟺}{8蟺r}+\frac{{渭}_{0}i蟺}{16蟺r}+\frac{{渭}_{0}i蟺}{12蟺r}=\frac{13{渭}_{0}i}{48r}$

Hence, the ranking of circuits according to the magnitude of the net magnetic field at the center is,$a>c>B$

Therefore, the magnetic field at the center of arc depends inversely on its radius.