The magnetic field due to wire from infinity to 0 that expands isotropically on the plane $z=0$

Question

An intensity current I descends down the $z$-axis from $z = \infty$ to $z = 0$, where it spreads out in an isotropic way on the plane $z = 0$. Compute the magnetic field.

To begin this problem I decided to break it into two problems. One where I solve for the magnetic field due to the wire and where I solve for the magnetic field due to the plane and add the two together for the total magnetic field. The magnetic field due to the wire is just half that of the magnetic field due to the infinite wire. So that $\vec{B} = \frac{\mu_0 I}{4\pi s} \hat{\phi}$, but I'm not sure if this works in all of space. For instnace, below $z=0$ the field would be a bit weirder I think.

The second thing I'm not sure about is how I compute the field due to the plane. I'm not sure what kind of loop to choose to make the problem easier. I think that the current surface density would be $\vec{K} = \frac{I}{r^2} \hat{r}$ so that it is isotropic and falls off as $r^2$

Answer 1

First of all: the magnetic field due to the semi-infinite wire is not directly half that of the infinite wire. You need to solve it again using Biot-Savart law. The problem is not symmetric like that of the infinite wire, and the magnetic field will change with the z location of the point. Second: the isotropic spread of the current I in the horizontal plane should be taken on the circumference of any circle with radius r. Then, the surface current density is $I/(2\pi r).$ The distribution isotropically in the plane is not like that in space. To compute the field due to the plane, use Biot-Savart law. The problem is not symmetric to choose an Amperian loop.

Answer 2

One method is as follows. There is a formula for the magnetic field of a line segment, e.g. example 5.5 in Griffiths, Introduction to Electrodynamics. The answer can then be obtained by integrating. The result will be that the magnetic field for $z >0$ is equal to the magnetic field of an infinite wire (!) and the magnetic field for $z<0$ is zero (!). I wrote a bit more detail here