Radiation Term's Relationship to r & \lambda

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Homework Statement

What is the condition for the radiation term in the expression for the field B to predominate? Express you answer as a relationship between distance r and wavelength $\lambda$.

B(r)=$\frac{\mu_{0}I_{0}\delta l}{4\pi}$sinθ ($\frac{-i\omega}{rc}$+$\frac{1}{r^{2}}$)exp[i(kr-$\omega$t)]$\phi$-hat

The Attempt at a Solution

r>>$\lambda$?
But how was this result obtained? If r>>c/omega, r>>c/(2$\pi$f), r>>$\lambda$/(2$\pi$), then how was the 2$\pi$ got rid of?

If r>>c/$\omega$, why is this the case?

Please help.

 

[mark726]

The condition for the radiation term to predominate in the expression for the magnetic field, B, is when the distance, r, is much larger than the wavelength, \lambda. This can be understood by examining the terms in the expression for B and their dependence on r and \lambda.

The term sin\theta is a constant and does not depend on r or \lambda. The term -(i\omega/rc) represents the phase of the wave and is inversely proportional to r. This means that as r increases, this term decreases and becomes negligible compared to the other terms. Similarly, the term 1/r^2 decreases as r increases.

The term exp[i(kr-\omegat)] represents the amplitude of the wave and is inversely proportional to r, while directly proportional to \lambda. This means that as r increases, this term decreases, but not as quickly as the previous terms.

Now, if we consider the case where r>>\lambda, we can see that the terms involving r will become very small compared to the terms involving \lambda. This is because when r is much larger than \lambda, the decrease in the terms involving r will be much greater than the decrease in the terms involving \lambda.

To address your question about the 2\pi, it is important to note that the expression for B is a complex quantity. The 2\pi is not "got rid of" but rather is included in the imaginary component of the expression. This is because the wave is described by a complex exponential function, which has a periodicity of 2\pi.

In summary, the condition for the radiation term to predominate in the expression for the magnetic field is when r>>\lambda. This is because as r increases, the terms involving r decrease faster than the terms involving \lambda, making them negligible. The 2\pi is not "got rid of" but is included in the imaginary component of the expression.

I hope this helps clarify the condition for the radiation term to predominate. Let me know if you have any further questions.



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