Question

Figure 29-24 shows three circuits, each consisting of two radial lengths and two concentric circular arcs, one of radius rand the other of radius R>r. The circuits have the same current through them and the same angle between the two radial lengths. Rank the circuits according to the magnitude of the net magnetic field at the center, greatest first

Short answer

The circuits according to the magnitude of the net magnetic field at the center, greatest first is c>a>b.

Step-by-step solution

Step 1: Given

• Figure of three circuits in which R > r.
• Same current is flowing through each circuit and the angle between two radial lengths is the same.

Determining the concept

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. Rank the givencircuits according to the magnitude of the magnetic field at the center.

The expression for magnetic flux density is given by,

$B=\frac{{\mu }_{0}if}{4\pi R}$

Where, B is magnetic field, R is radius, iis current,𝛍is permeability.

Determining the rank of circuits according to the magnitude of the net magnetic field at the center, greatest first.

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

$B=\frac{{\mu }_{0}if}{4\pi R}$

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

$B=\frac{{\mu }_{0}i\left(2\pi -f\right)}{4\pi R}+\frac{{\mu }_{0}if}{4\pi r}$

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

$B=\frac{{\mu }_{0}if}{4\pi r}-\frac{{\mu }_{0}if}{4\pi R}$

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

$B=\frac{{\mu }_{0}i\left(2\pi -f\right)}{4\pi r}+\frac{{\mu }_{0}if}{4\pi R}$

Since, $R>randf\ll 2\pi -f$, the ranking of circuits according to the magnitude of the net magnetic field at the center is,

c > a > b

Hence, the circuits according to the magnitude of the net magnetic field at the center, greatest first is c > a > b.

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