Chapter 29, Problem 1Q

To determine

To rank:

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

Answer to Problem 1Q

Solution:

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

Explanation of Solution

1) Concept:

We can find the net magnetic field at the center of each circuit using the formula for magnetic field at the center of circular loop. Comparing them we can rank the given circuits according to the magnitude of the magnetic field at the center.

2) Formulae:

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

3) Given:

i) Figure 29.24 of three circuits, each consisting of two radial lengths, and two concentric circular arcs, one with radius r, and the other R > r.

ii) Same current is flowing through each circuit and angle between two radial lengths is same.

4) Calculations:

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

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

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

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

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

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

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

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

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

$c>a>b$

Conclusion:

The magnetic field at the center of an arc depends inversely on its radius.