Constant magnetic field > The magnetic field from a current distributed over a surface of space > Problem 9.3.15

Statement

$9.3.15.$ The circulation of the induction of a constant magnetic field through a closed circuit in a vacuum is equal
to the current through the surface bounded by this circuit, multiplied by $µ0$. Please provide examples
that support this law. Use it to solve the following problems:

a. An infinitely long straight wire of radius $r$ flows current $I$. The current is distributed evenly over the
wire cross-section. Find the magnetic field induction inside and outside the wire.

b. A long, wide busbar with a cross dimension of a flows a current evenly distributed over the crosssection of the conductor. Current density $j$. How does the magnetic field induction depend on the distance $x$ to the median plane of the tire?

Solution

а)
Let's take a circular contour of radius $R$, when $R>r$ all the current enters the contour:
$$B\cdot 2\pi R = \mu_0 I$$
$$B=\frac{\mu_0 I}{2\pi R}$$
with $R < r$：
$$B\cdot 2\pi R= \mu_0 I \frac{R^2}{r^2}$$
$$B=\frac{\mu_0 I R}{2\pi r^2}$$
b) Let's consider a rectangular contour with dimensions
$L$ and $2x$, with $x < a/2$:
$$B \cdot 2L = \mu_0 \cdot 2jxL$$
$$B = \mu_0 jx$$
with $x >= a/2$:
$$B = \frac{\mu_0ja}{2}$$

Savchenko has a typo in the inequality sign

Answer

а. $B = \frac{\mu_0Ix}{2πr^2}$, $0 < x < r$; $B = \frac{\mu_0I}{2πx}$, $x > r$.

b. $B =\mu_0 xj$, $x < a/2$; $B =\frac{\mu_0 aj}{2}$, $x >= a/2$.