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## Homework Statement

I have a long EM question in which there is a Hertzian dipole at a point (0,0,-100), (unknown orientation) and I am told the equation of the physical magnetic field detected 100m away at the origin of Cartesian coordinates. $$(B_0 \sin (2 \pi f t)\mathbf{e}_x$$, and $$B_0 = 0.1 \mu T$$, $$f=30 MHz$$.

I have to deduce the possible orientations of the Hertzian dipole and explain qualitatively why the plane wave approximation is valid at +/- 1m in any direction from the origin.

## Homework Equations

Maxwell's equations.

## The Attempt at a Solution

On the possible directions of the Hertzian dipole: using the Biot-Savart law, I think I can show that the vector product of the current element and the position vector have to produce positive e_x, which is the detected field. So I'm thinking that the dipole has to be pointing in the -e_y direction, because:

$$\mathbf{e}_{?} \times -\mathbf{e}_z = \mathbf{e}_x$$ gives me the -y direction, although two problems: I don't know how to write that argument mathematically (can't divide by a vector) and it only gives me one possible direction, where the question suggests there are more. The question also has a lot of marks attributed to it, and I'm wondering whether I need to talk about possible angular dependence of the dipole.

On the approximation part: I'm guessing that the plane wave approximation is only valid under certain conditions (maybe when the distance is much larger than the wavelength), but I'm unsure as to exactly how this comes in and whether or not the frequency plays a part.

I would gratefully appreciate any pointers.

Best wishes

P.S. A thought just occurred to me that the Hertzian dipole must have a -y component to explain the e_x component of the magnetic field, but any z component could also exist and would simply disappear under the vector product operation...that gives me a new range of direction possibilities...

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