Constant magnetic field > The magnetic field from a current distributed over a surface of space > Problem 9.3.17

Statement

$9.3.17.$ a. The current $I$ goes along a long straight wire perpendicular to the conducting plane and spreads
along it. Determine the magnetic field distribution.

b. A long wire with current $I$ intersects the conducting plane in a direction perpendicular to it. The
current flowing to the plane is $I'$ . Determine the distribution of the magnetic field in this system.

c. The coaxial cable enters the spherical plane as shown in the figure. Find the magnetic field induction
in the entire space.

Solution

а)
Let's surround the wire with rings of radius $2\pi x$, above the plane they carry current I and induction $$B = \frac{\mu_0 I}{2\pi x}$$, below the plane there is no current, therefore the induction is zero.

b) Similarly, over the plane the induction $$B = \frac{\mu_0 I}{2\pi x}$$ under: $$B = \frac{\mu_0 (I-I')}{2\pi x}$$

c) Using the same type of circuit, we'll analyze it. Inside the circuit, the current only enters. The answer is $$B = \frac{\mu_0 I}{2\pi x}$$
Let's take a wider circuit. The current enters it and the same amount exits $\Longrightarrow$ outside $$B = 0$$

Answer

а.
Above the plane $B = \frac{\mu_0I}{2πx}$, the magnetic field lines coincide with the field lines of an infinite straight wire; below the plane $B = 0$

b.
Above the plane $B = \frac{\mu_0I}{2πx}$, below the plane $B = \frac{\mu_0(I − I′)}{2πx}$.

c.
Inside the cable $B = \frac{\mu_0I}{2πx}$, outside the cable $B = 0$.