One-dimensional lattice (electrostatics)

[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"?

Thanks in advance.

Homework Equations

The Attempt at a Solution

 

[Hootenanny]

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.

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?

 

[asi123]

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.

 

[Hootenanny]

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 q_i and q_j are the charge of the i^th and j^th particle respectively. And r_ij 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 r_ij in terms of b?

 

[asi123]

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 q_i and q_j are the charge of the i^th and j^th particle respectively. And r_ij 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 r_ij 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