Electric field due to a point dipole

In summary, in the far field, the electric field due to a point dipole oscillating along the z axis can be described by the equation E = \frac{d}{4\pi\epsilon_0}\frac{k^2\sin\theta}{r}\exp i\left(kr-\omega t\right), where d is the dipole moment, k is the wave number, \theta is the angle made with the z axis, r is the radial distance, \omega is the angular frequency, and t is the time. The equation suggests a doughnut-shaped electric field, with maximum intensity at \theta=\pi/2 in the x-y plane and zero intensity along the z axis. However, taking cross sections
  • #1
Hypochondriac
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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.
 
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  • #2
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 x-y plane. I think if you take a cross section of the donut on the z-y or z-x planes you'll find the lobes you are picturing. However, the lobes are centered in the x-y plane, not along z.

For reference, see Griffith's Eq. 11.18 and Figure 11.4
 

1. What is a point dipole?

A point dipole is a simple model used to describe the behavior of electric charges. It consists of two equal and opposite point charges separated by a small distance. This small distance is known as the dipole moment.

2. How is the electric field calculated for a point dipole?

The electric field at a point due to a point dipole is calculated by summing the electric field contributions from each point charge. The formula for the electric field at a distance r from the dipole is E = (1/4πε0) * [(q/r2) - (q/r3)] * d, where q is the magnitude of the point charges and d is the dipole moment.

3. What is the direction of the electric field for a point dipole?

The direction of the electric field for a point dipole is determined by the orientation of the dipole moment. The electric field lines point away from the positive charge and towards the negative charge, forming a dipole pattern.

4. How does the strength of the electric field change with distance from a point dipole?

The strength of the electric field due to a point dipole decreases with distance from the dipole. As the distance increases, the electric field decreases in proportion to 1/r2, where r is the distance from the dipole.

5. What are the applications of understanding the electric field due to a point dipole?

Understanding the electric field due to a point dipole is important in various fields such as electromagnetism, chemistry, and biology. It is used to describe the behavior of polar molecules, the interaction between charged particles, and the behavior of electric dipoles in an electric field.

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