# Dipoles in 2D and 'dipolar line'

I am currently studying the textbook Surface Enhanced Raman Spectroscopy -- Analytical, Biophysical, and Life Science Applications by Sebastian Schlücker, Wolfgang Kiefer. Chapter 1.2.3.2 Localized Surface Plasmon Resonances of the Cylinder says the following:

The exact solution of a 2D cylinder with dielectric function $$\epsilon(\lambda)$$ turns out to be analytically tractable. It turns out that boundary conditions on the surface of the cylinder can be exactly satisfied by considering the superposition of the external field $$E$$ with an induced dipole centred at the origin, as depicted in Figure 1.3b.

Figure 1.3 (a) A cylindrical metallic object is impinged by an electromagnetic wave coming from the side with wavevector $$k_i$$ and polarization $$E_i$$ (on the plane of the page). (b) When the object is small compared to the wavelength ($$\le 10-20 nm$$), it is possible to gain some insight into the situation by solving the problem electrostatically (i.e. in a constant electric field in the direction of the polarization and with no wavevector present). The boundary conditions at different wavelengths ($$\lambda = 2\pi c / \omega$$) are still specified by ($$\lambda$$), and can be fulfilled exactly at the surface of the cylinder by considering the superposition of an induced dipole and the external field (Equation 1.4).

As far as points outside the cylinder is concerned then, the electric field looks like the superposition of this dipole ($$p$$) at the origin and the external field. We shall not go into the details of the solution of the electrostatic problem to keep the mathematical aspects to a bare minimum, but rather only mention that the magnitude of the induced dipole satisfies the boundary conditions and solves the electrostatic problem is proportional to

$$p \propto \dfrac{\epsilon(\lambda) - \epsilon_M}{\epsilon(\lambda) + \epsilon_M} \tag{1.4}$$

Dipoles in 2D are more complicated than standard dipoles in 3D. Note that it should be described more formally as a 'dipolar line' (in the direction perpendicular to the page as shown in Figure 1.3) rather than a dipole.

$$\epsilon(\lambda)$$ is the dielectric function, and $$\epsilon_M$$ is the dielectric medium (I'm guessing this means that it is the dielectric constant for the medium).

It is this last part the I am curious about:

Note that it should be described more formally as a 'dipolar line' (in the direction perpendicular to the page as shown in Figure 1.3) rather than a dipole.

It wasn't clear to me what was meant by this 'dipolar line' concept. I would greatly appreciate it if people would please take the time to carefully explain this to a novice such as myself.

The topic under discussion here is plasma resonance in a cylindrical structure. The geometry here is cylindrical. The exciting field is perpendicular to the axis of the cylinder, and the polarization is also perpendicular to the cylinder. He is saying that the dipole moment that the field induces can be considered to live on the axis ... hence a "dipolar line".

• Thanks for the answer. So living on this axis mathsteacher.com.au/year9/ch14_measurement/14_curved/… ? May 13, 2020 at 16:47
• The image I see from that link is extremely distorted. It looks like random shapes and words. May 13, 2020 at 16:56
• It's the second image here mathsteacher.com.au/year9/ch14_measurement/14_curved/… under "Curved Surface Area of a Cylinder". The axis is labelled in the cylinder. May 13, 2020 at 16:59
• Yes, the axis of the cylinder. May 13, 2020 at 17:01
• Thanks for the clarification. I will award the bounty when it allows me to do so. May 13, 2020 at 17:03