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Chapter 8: Problem 2

Consider an electric dipole consisting of a charge \(-e\) oscillating sinusoidally in position about a stationary charge \(+e\). Show that the instantaneous total power radiated can be written in the form $$ \frac{d W}{d t}=\frac{e^{2}[a]^{2}}{6 \pi \epsilon_{0} c^{3}} $$ where \([a]=a_{0} e^{i(\omega t-k n)}\) is the instantaneous (retarded) acceleration of the moving charge Since this result does not depend upon the oscillator frequency, and since by Fourier analysis, an arbitrary motion can be described by superposing many sinusoidal motions of proper frequency, amplitude, and phase, this rate-of-radiation formula has general validity for any accelerated charge (in the nonrelativistic limit \(\diamond \ll c\) ).

Short Answer

Expert verified
To show that the instantaneous total power radiated by an electric dipole with a sinusoidally oscillating charge and a stationary charge can be written in the given form, we first analyzed the motion of the oscillating charge and found its acceleration. Then, we applied Larmor's formula to compute the power radiated by the charge. By expressing the acceleration in the retarded form, we simplified the expression for the power and found the real part which represents the measurable power radiated. This expression is general and valid for any accelerated charge within the nonrelativistic limit.

Step by step solution

01

Analyze the motion of the oscillating charge

Assume the charge \(-e\) oscillates along the x-axis. Its position as a function of time is given by: $$x = x_0 \cos(\omega t).$$ Differentiate the position function twice with respect to time to obtain the acceleration: $$a(t) = -\omega^2 x_0 \cos(\omega t).$$ Now, we can express the acceleration in the given retarded form: $$[a] = a_0 e^{i(\omega t - k n)},$$ where \(a_0=-\omega^2 x_0\).
02

Apply Larmor's formula

Larmor's formula expresses the power radiated by an accelerating charged particle: $$\frac{dW}{dt} = \frac{e^2 a^2}{6 \pi \epsilon_0 c^3}.$$ Plug the expression for acceleration \([a]\) into this formula: $$\frac{dW}{dt} = \frac{e^2 a_0^2 e^{2i(\omega t - k n)}}{6 \pi \epsilon_0 c^3}.$$
03

Simplify the expression and find the power radiated

We can simplify the expression by finding the real part of the power, as only the real part represents the measurable power radiated. $$\frac{dW}{dt} = \frac{e^2 a_0^2 \cos^2(\omega t - k n)}{6 \pi \epsilon_0 c^3}.$$ Notice that this expression does not depend on the oscillator frequency \(\omega\). By Fourier analysis, arbitrary motion can be described by superposing sinusoidal motions of various frequencies, amplitudes, and phases. Therefore, this formula has general validity for any accelerated charge, within the nonrelativistic limit (\(v \ll c\)).

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Most popular questions from this chapter

(a) For waves varying sinusoidally with time as \(e^{j w t}\), show that the conductivity can be eliminated from (8.5.7) and (8.5.8) by substituting for the relative permittivity the complex quantity $$ \vec{k}_{e}=\kappa_{q}-j \frac{g}{\omega t_{0}} . $$ Then all electromagnetic properties of the medium are contained in only two constants, \(\vec{x}_{e}\) and \(\kappa_{m}\). (b) When currents flow nonuniformly in space, it is possible that a net charge density Pfres builds up at certain locations, Show that the complex permittivity formalism of part (a) not only eliminates the \(\mathbf{J}_{\text {freo }}\) term in Maxwell's equation \((8.2 .20)\) but also eliminates the piros term in \((8.2 .17)\). (c) As an alternative to the formalism of part ( \(a\) ), show that the relative permittivity can be disregarded, i.e., set equal to unity, by introducing the complex conductivity $$ \ddot{g}=g+j \cot \theta\left(\kappa_{c}-1\right) \text {. } $$ In this case, the properties of the medium are specified by the two constants \(g{g}\) and \(\kappa_{w}\) -

Substitute (8.9.3) in (8.9.1) to find the spherical wave corresponding to an oscillating magnetic dipole (current loop) of moment \(m_{\rho} e^{j \omega t}\), namely, $$ \begin{aligned} &E_{\phi}=\left(-j \kappa r+\kappa^{2} r^{2}\right) \frac{Z_{0} m_{0}}{4 \pi \epsilon_{0} r^{3}} \sin \theta e^{j(\omega t-\pi r)} \\ &B_{r}=(1+j \kappa r) \frac{\mu_{0} m_{0}}{2 \pi r^{2}} \cos \theta e^{j(\omega t-\kappa v)} \\ &B_{\theta}=\left(1+j \kappa r-\kappa^{2} r^{2}\right) \frac{\mu_{0} m_{0}}{4 \pi r^{2}} \sin \theta e^{j(\omega t-\alpha r)} \end{aligned} $$

The text following (8.2.10) refers to low-frequency (or dc) laboratory measurements of \(\epsilon_{0}\) and \(\mu_{0}\). How could you determine these constants? What logical chain of definitions and calibrations would be needed?

If oscillatory fields are represented by \(\mathbf{E}=\underline{\mathbf{E}}_{0} c^{j \omega t}\) and \(\mathbf{H}=\breve{\mathbf{H}}_{0} e^{j \omega t}\), using the realpart convention, show that the (real) Poynting vector is given by $$ s=\frac{1}{2}\left(\mathbf{E} \times \mathbf{H}^{*}+\mathbf{E}^{*} \times \mathbf{H}\right)=\operatorname{Re}\left(\underline{\mathbf{E}} \times \check{\mathbf{H}}^{*}\right)=\operatorname{Re}\left(\breve{\mathbf{E}}^{*} \times \breve{\mathbf{H}}\right), $$ where the asterisk denotes complex conjugate. Also, show that the time-average Poynting vector is

Develop Poynting's theorem for the general material medium of relative permittivity \(\kappa_{\&}\) and permeability \(\kappa_{m}\) introduced in Prob. \(8.2 .2\); i.e., substitute the Maxwell curl equations \((8.2 .18)\) and \((8.2 .20)\) in the expansion of \(\nabla \cdot(\mathbf{E} \times \mathbf{H})\) to obtain $$ \oint_{S}(\mathbf{E} \times \mathbf{H}) \cdot d \mathbf{S}+\int_{V}\left(\mathbf{E} \cdot \frac{\partial \mathbf{D}}{\partial t}+\mathbf{H} \cdot \frac{\partial \mathbf{B}}{\partial t}\right) d v+\int_{V} \mathbf{E} \cdot \mathbf{J} d v=0 $$ from which it follows that the Poynting vector is $$ s=\mathbf{E} \times \mathbf{H} $$ and the energy density is $$ \begin{aligned} \left(W_{1}\right)_{\text {leld }} &=\frac{1}{2} \mathbf{E} \cdot \mathbf{D}+\frac{1}{2} \mathbf{H} \cdot \mathbf{B} \\ &=\frac{1}{2} \kappa_{\ell} \epsilon_{0} E^{2}+\frac{1}{2 \kappa_{\mathrm{m}} \mu_{0}} B^{2} \end{aligned} $$ What restrictions on \(\kappa_{\&}\) and \(\kappa_{m}\) are necessary to obtain \((8.4 .24)\) ?
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