# Induced scalar electric potential

• nykon
In summary: For me it's not a big deal, I just copy and paste it. In summary, the energy loss per unit time W is due to resistive dissipation.
nykon
Hi Forum!

I have got a question about the induced scalar potential. I will present the problem from beginning.

Lets say we have a Poisson's equation in form:

$\epsilon \nabla^2 \phi = -4\pi \varrho(r,t)$

where $\epsilon$ is the dielectric constant. By use of the Fourier transform:

$f(r,t) = \int \frac{d^3q}{(2\pi)^3} \int_{-\infty}^{\infty} \frac{d\omega}{2\pi} e^{i(qr-\omega t)}f(q, \omega)$,

(where $q$ is the momentum, and $\omega$ is the energy) one can write:

$\phi_{q, \omega}= \frac{4\pi \varrho_{q, \omega}}{q^2 \epsilon}$

where $\varrho_{q, \omega} = 2\pi Z \delta(\omega - q v)$, v is the velocity of the incident ion.

Now if we assume that the incident ion is moving through the electron gas we can write the induced scalar potential in form:

$\phi^{induced}_{q, \omega}= \frac{8\pi^2 Z}{q^2} \delta(\omega - q v)(\frac{1}{\epsilon} - 1)$

Now the energy loss per unit time W is:

$W = - Z v E^{induced}, \qquad E^{induced} = -\nabla \phi^{induced}$

the final result is:

$W = \int \frac{d^3q}{(2\pi)^3} \int_{-\infty}^{\infty} \frac{d\omega}{2\pi} 2 \omega Z Im[-\phi^{induced}]$,

My question is: why I need the imaginary part in the final result of the induced scalar potential? The result is taken from the "dynamic screening of ions in condensed matter" written by Echenique, Flores and Ritschie. I just do not understand the last formula. I will be gratefull for any tip or advise.

nykon

The dielectric constant is complex, right? The ion is moving through an electron gas, which is a conducting medium. And W, the energy loss per unit time, is due to resistive dissipation. Anyway, I'm thinking that the previous line where W is given in terms of E should have an imaginary part on it also.

Thats why I am confused. I really don't know why in the "final" equation the $Im(-\phi_{q,w})$ appears. Why we "cut" the Re part? These textbook is available online, but I guess one has to pay for it.

Another publication where one can find these formulas is: "Interaction of Slow Ions with Matter" written by Echenique, Nagy and Arnau http://onlinelibrary.wiley.com/doi/10.1002/qua.560360854/abstract?systemMessage=Wiley+Online+Library+will+be+disrupted+9+July+from+10-12+BST+for+monthly+maintenance"

I have found a loot of publications from 1955 to 2002 where people use just the same notation, "way of thinking" and even steps are similar.

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## 1. What is an induced scalar electric potential?

An induced scalar electric potential refers to the electric potential that is created in a material due to the presence of an external electric field. This potential is caused by the displacement of charged particles within the material.

## 2. How is induced scalar electric potential different from regular electric potential?

Regular electric potential is created by stationary charges, while induced scalar electric potential is caused by the movement of charges within a material. Additionally, induced scalar electric potential is dependent on the strength and direction of the external electric field, while regular electric potential is not.

## 3. Can induced scalar electric potential be measured?

Yes, induced scalar electric potential can be measured using various instruments such as voltmeters or oscilloscopes. The magnitude of the induced potential can be determined by the strength of the external electric field and the properties of the material.

## 4. What are some applications of induced scalar electric potential?

One common application of induced scalar electric potential is in the operation of electronic devices, where it is used to control the flow of electric current. It is also used in the production of electricity through electromagnetic induction, as well as in medical procedures such as electrocardiograms.

## 5. How does induced scalar electric potential relate to Faraday's law?

Faraday's law states that a changing magnetic field can induce an electric field. This induced electric field can then cause a scalar electric potential in a material, as the movement of charges within the material creates the potential. Therefore, induced scalar electric potential is closely related to Faraday's law of electromagnetic induction.

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