An electromagnetic signal is generated by a Hertzian dipole located at a point P

Question

An electromagnetic signal is generated by a Hertzian dipole located at a point P, which has the position vector r = -(100m)e_z. The signal is detected by a small wire loop located at the origin. Apart from the dipole and the loop, the nearby space is empty. Experimentation reveals that the detected signal is induced by a changing magnetic field: B_phys (t) B_0 sin (2pi f t) e_x, where B_0 = 0.1 mu T and f = 30 MHz. (c) Show that the given physical magnetic field at the loop. B_phys(t), is consistent with a monochromatic plane wave solution of Maxwell's equations given by B = i B_0 exp[i(kz - omega t)] e_x.

Explanation / Answer

This is straight forward answer

Since any solution of the wave equation can be expressed as a sum (or integral) over sinusoidal functions (that’s Fourier analysis)

Considering waves that consist of only a single frequency 'w' that travel in the +z direction and have no dependence on x or y, we can write the solutions as

Such a wave is called monochromatic (“one colour”) because it contains only one frequency (and hence, for visible light, only one colour) and plane because the wave is constant over any plane perpendicular to the direction of propagation.

We can now apply Maxwell’s equations to these solutions. First, an observation about the complex notation. For the fields above, the real parts depend on space and time through a term cos(kz-wt) and the imaginary parts through a term sin(kz-wt). which will be similar representation as the above given physical magnetic fiels at the loop and hence in consistence.

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