# One-dimensional lattice (electrostatics)

• asi123
In summary: The potential caused by two nearest charges:V1 = -kq^2/b * 2ktop(multiply by 2 because there are two nearest charges)Then the potential caused by two next charges:V2 = kq^2/(2b) *2ktop(it's positive, because they have same sign)And then:V3 = -kq^2/(3b) *2ktopV4 = kq^2/(4b) *2ktop...Sum all of these potential energy,Vtot = V1 + V2 + V3 + ...+ Vtot = 2*kq
asi123

## Homework Statement

Hey guys.
So, I got this question in the pic.
First of all, I drew what I think to be a one-dimensional lattice (in the green box) but I'm not sure, is it right?
Second of all, I don't really understand the question, I mean I know that a potential energy of charge q is V(r) = kq/r when you say of curse that v(infinity) = 0 but what do they mean by a "potential energy of a single charge"?

## The Attempt at a Solution

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asi123 said:
First of all, I drew what I think to be a one-dimensional lattice (in the green box) but I'm not sure, is it right?
Your 1D lattice looks good to me.
asi123 said:
Second of all, I don't really understand the question, I mean I know that a potential energy of charge q is V(r) = kq/r when you say of curse that v(infinity) = 0 but what do they mean by a "potential energy of a single charge"?
You should be careful here, you have made a very common mistake. The potential of a point charge is given by the equation you quote. However, the potential energy is given by a different equation and corresponds to the work done moving a charge from infinity (or any other arbitrarily fixed point) to it's current location. For more information see here: http://hyperphysics.phy-astr.gsu.edu/Hbase/electric/elepe.html#c3

Do you follow?

Hootenanny said:
Your 1D lattice looks good to me.

You should be careful here, you have made a very common mistake. The potential of a point charge is given by the equation you quote. However, the potential energy is given by a different equation and corresponds to the work done moving a charge from infinity (or any other arbitrarily fixed point) to it's current location. For more information see here: http://hyperphysics.phy-astr.gsu.edu/Hbase/electric/elepe.html#c3

Do you follow?

Yeah I follow.
I know this kind of questions (how much energy does it take to build a sphere and such...)
However, I don't understand the question, is it, how much energy does it take to build this kind of lattice?

Thanks again.

asi123 said:
Yeah I follow.
I know this kind of questions (how much energy does it take to build a sphere and such...)
However, I don't understand the question, is it, how much energy does it take to build this kind of lattice?

Thanks again.
Yes you are correct, the potential energy of a system of charges is basically the energy required to build up the system (i.e. bring each charge from infinity to it's current position). This concept can be formalised as a sum (for N particles):

$$U = \kappa\sum_{\stackrel{i,j=1}{i\neq j}}^N \frac{q_iq_j}{\mathbf{r}_{ij}}$$

Where qi and qj are the charge of the ith and jth particle respectively. And rij is the relative position vector (or the separation distance in the 1D case) of the two particles. It is important to note that the sum excludes the case when the indices are equal.

In your case, we have an infinite lattice and hence an infinite sum. This is where the hint in the question comes in handy. Can you write rij in terms of b?

Hootenanny said:
Yes you are correct, the potential energy of a system of charges is basically the energy required to build up the system (i.e. bring each charge from infinity to it's current position). This concept can be formalised as a sum (for N particles):

$$U = \kappa\sum_{\stackrel{i,j=1}{i\neq j}}^N \frac{q_iq_j}{\mathbf{r}_{ij}}$$

Where qi and qj are the charge of the ith and jth particle respectively. And rij is the relative position vector (or the separation distance in the 1D case) of the two particles. It is important to note that the sum excludes the case when the indices are equal.

In your case, we have an infinite lattice and hence an infinite sum. This is where the hint in the question comes in handy. Can you write rij in terms of b?

Well, I was thinking about something like that:

The potential caused by two nearest charges:
V1 = -kq^2/b * 2
(multiply by 2 because there are two nearest charges)

Then the potential caused by two next charges:
V2 = kq^2/(2b) *2
(it's positive, because they have same sign)

And then:
V3 = -kq^2/(3b) *2
V4 = kq^2/(4b) *2
.
.
.

And sum all of these potential energy,
Vtot = V1 + V2 + V3 + ...
Vtot = 2*kq^2/b (-1+1/2-1/3+1/4-1/5+...)

Is this right?

Thanks a lot BTW

## 1. What is a one-dimensional lattice in terms of electrostatics?

A one-dimensional lattice in electrostatics refers to a series of equally spaced charged particles or ions arranged in a straight line. It is commonly used to model the behavior of ions in a crystal lattice or in other one-dimensional systems.

## 2. How is the electric potential of a one-dimensional lattice calculated?

The electric potential of a one-dimensional lattice can be calculated using the Coulomb's law, which states that the potential at a point due to a single charged particle is directly proportional to the magnitude of the charge and inversely proportional to the distance from the particle.

## 3. What is the significance of a one-dimensional lattice in electrostatics?

A one-dimensional lattice is important in electrostatics as it allows us to understand the behavior of charged particles in a simplified system. It can also be used to model more complex systems, such as a crystal lattice, and help us understand their behavior and properties.

## 4. How does the behavior of a one-dimensional lattice differ from a two-dimensional or three-dimensional lattice?

In a one-dimensional lattice, particles are only able to interact with their nearest neighbors, whereas in a two-dimensional or three-dimensional lattice, particles can interact with a larger number of neighbors. This results in different patterns of charge distribution and behavior of the particles in each type of lattice.

## 5. Can a one-dimensional lattice be used to model real-world systems?

Yes, a one-dimensional lattice can be used to model various real-world systems, such as the arrangement of atoms in a solid or the movement of ions in a channel. It provides a simplified yet accurate representation of these systems and helps us understand their properties and behavior.

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