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Electric field due to a point dipole 
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#1
Feb411, 02:55 PM

P: 35

Let's say I have a point dipole (as an approximation for an atom) at the origin and it oscillates in the [tex]z[/tex] axis. The (theta component of the) electric field due to this dipole in the far field will be
[tex]E = \frac{d}{4\pi\epsilon_0}\frac{k^2\sin\theta}{r}\exp i\left(kr\omega t\right)[/tex] where [tex]d[/tex] is the dipole moment, [tex]k=2\pi/\lambda[/tex], [tex]\theta[/tex] is the angle made with the [tex]z[/tex] axis, [tex]r[/tex] is the radial distance, [tex]\omega[/tex] is the angular frequency of the oscillation and [tex]t[/tex] is the time. Due to the zero [tex]\phi[/tex] dependance, i.e. the angle in the equatorial plane, there is a cylindrical symmetry. Instinct tells me that I should I have two lobes of electric field, one in the [tex]+z[/tex] direction and one in [tex]z[/tex], which will oscillate, alternatively between positive and negative. However the equation I quoted implies a doughnut shaped electric field. For a given [tex]r[/tex], [tex]E[/tex] increases as [tex]\theta[/tex] goes from 0 to [tex]\pi/2[/tex], then decreases from [tex]\pi/2[/tex] to [tex]\pi[/tex]. I.e. I think the equation should have a [tex]\cos\theta[/tex] in it instead of a [tex]\sin\theta[/tex] Where have I gone wrong? I think it's my definition of [tex]\theta[/tex], however it is always defined from the [tex]z[/tex] axis. PS. I'm pretty sure the equation is in Jackson. 


#2
Feb411, 04:23 PM

HW Helper
P: 2,685

This is the correct form for a point dipole oscillating along z. The intensity profile for a point dipole source is indeed like a donut.
Since the dipole is oscillating along z, there is 0 intensity at [itex]\theta=0[/itex], and the intensity is maximum [itex]\theta=\pi /2[/itex] in the xy plane. I think if you take a cross section of the donut on the zy or zx planes you'll find the lobes you are picturing. However, the lobes are centered in the xy plane, not along z. For reference, see Griffith's Eq. 11.18 and Figure 11.4 


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