Wavelength of an Electromagnetic wave

[rohanprabhu]

I tried to see how we go about *creating* and electromagnetic wave. To do that, I take a charge 'Q' rotating in a circle of 'radius' r in the y-z plane, with it's center as the origin with an angular velocity $\omega$. The electric field at a point (x, y, z) is given by:

$$E(x, y, z) = E_x(x, y, z) + E_y(x, y, z) + E_z(x, y, z)$$

where:

$$E_x(x, y, z) = \frac{Q(9 \times 10^{9})x}{\left(r^2+x^2+y^2+z^2-2 r (y \text{Cos}[t \omega ]+z \text{Sin}[t \omega ])\right)^3}\hat{i}$$

$$E_y(x, y, z) = \frac{Q(9 \times 10^{9})(y<br /> - r\text{Cos}[t \omega ])}{\left(r^2+x^2+y^2+z^2-2 r (y \text{Cos}[t \omega ]+z \text{Sin}[t \omega ])\right)^3} \hat{j}$$

$$E_z(x, y, z) = \frac{Q(9 \times 10^9)(y - r\text{Sin}[t \omega ])}{\left(r^2+x^2+y^2+z^2-2 r (y \text{Cos}[t \omega ]+z \text{Sin}[t \omega ])\right)^3} \hat{k}$$

Is this an electromagnetic wave. I think it is because well.. since the charge retraces it's path every $$t = \frac{2\pi}{\omega}$$.. so the electric field at any point will vary periodically as a function of time.

How do I find it' wavelength and frequency? Is $$f = \frac{1}{T} = \frac{\omega}{2\pi}$$ correct for frequency?

Also.. how do i find the generated magnetic field? And assuming that the wavelength of this wave comes out to be something within the visible range of light.. will this moving charge cause visible radiation?

 

[pam]

You have just tried to use the electrostatic E.
Generating an EM wave is much more complicated, involving the retarded time and Maxwell's equations. You have to read the EM radiation chapter in a more advanced text.

 

[rohanprabhu]

You have just tried to use the electrostatic E.
Generating an EM wave is much more complicated, involving the retarded time and Maxwell's equations. You have to read the EM radiation chapter in a more advanced text.

any recommendations?

Also.. Maxwell did say that a changing electric field causes a changing magnetic field. In that case, the electric field which i presented, should also create a magnetic field. So, basically i have a time changing vector field consisting of both and electric field and a magnetic field.. so why is it not an electromagnetic wave?

 

[jtbell]

See for example chapter 9 of Griffiths's "Introduction to Electrodynamics."

The first example most books do is radiation from an oscillating dipole: a positive and negative charge "flip-flopping" back and forth in simple harmonic motion. A Google search on "dipole radiation" or something similar might turn up some lecture notes.

 

[monish]

Dipole radiation is a pretty hard case to analyze. A much simpler case for radiation is an infinite sheet with oscillating current. If the current density is 1 amp/meter (yes, those are the units for sheet current, not amps/meter squared!) then you create a field at right angles of 377 volts/meter and an associated power of 377 watts flowing away from the sheet.