9.3.15 | revision #18476 | Savchenko Solutions

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