Statistics of charged beads in electric field

[Beer-monster]

Homework Statement

Consider a string of N beads connected by N-1 rigid rods of length l. The system is considered as one-dimensional with rods only being aligned either up or down. The string is immersed in a fluid at temperature T and first bead is fixed at the origin y=0. A constant electric field is applied in the positive y direction.

What is the average length of the string if only the last bead holds a charge q.

Homework Equations

$$\sum\frac{e^{-\beta E}}{Z}$$

The Attempt at a Solution

I'm a little unsure about how to consider the effects of the charged bead in the field.

My thoughts in general were to consider the rods in two states, either aligned with or against the field. If aligned with the field the energy is negative $-eEl$. If aligned with the field the energy is positive $+eEl$. By considering these states I hope to determine the probability of a rod being aligned with/against field and from that determine the expectation value of the length.

One one hand if only the end bead is charged only that bead will show a preferential alignment and the rest will be random.

On the other hand; if the Nth bead feels a force due to the applied field it will also apply a force to the N-1th bead, which will apply the force to the N-2th bead etc. In that way the work will be applied to all of the rods so the energy shift will apply to all beads.

Which is right? Or am I completely off-base?

 

[TSny]

For the partition function, you need an expression for the energy of the system for each possible state of the system. A state of the system is a specification of the orientations of the individual rods. The energy of the system is equal to the electrical potential energy of the charged bead and that’s determined by the charged bead’s y coordinate, y_N.

If you knew the orientations of all the rods, how could you calculate y_N? It might be helpful to introduce a variable for each rod to denote it’s orientation, say s_k for the k^th rod. Let s_k = 1 or -1 for the rod being parallel or antiparallel to the field, respectively. Can you express y_N in terms of the s_k variables and the length of each rod?

How would you express the electrical potential energy of the system in terms of y_N?

 

[Beer-monster]

If we define Sk as +/-1 depending on if the rods are aligned with the field the length of the string would be:

$$y_{N}=\sum_{N} S_{k}l$$ where l is the length of a rod.

In the absence of an initial coordinate for the Nth bead I would express the potential energy of due to the field as $\epsilon = q_{N}Ey_{N}$

Thus the partition function per rod would be

$$Z = \sum^{+1}_{S_{k}=-1} e^{\beta q_{N}E\sum_{N} S_{k}l}$$

Using this I can determine the probability and thus the average number of a rod being aligned with or against the field and sum to make the average length.

 

[TSny]

If we define Sk as +/-1 depending on if the rods are aligned with the field the length of the string would be:

$$y_{N}=\sum_{N} S_{k}l$$ where l is the length of a rod.

Yes. The sum is over k from 1 to N-1?

In the absence of an initial coordinate for the Nth bead I would express the potential energy of due to the field as $\epsilon = q_{N}Ey_{N}$

The potential energy of a (positive) charge decreases as the charge moves in the direction of the field. So, the sign is not correct for the energy.

Thus the partition function per rod would be

$$Z = \sum^{+1}_{S_{k}=-1} e^{\beta q_{N}E\sum_{N} S_{k}l}$$

Here, the sign of the argument of the exponential is correct!

Using this I can determine the probability and thus the average number of a rod being aligned with or against the field and sum to make the average length.

Yes.

 

[Nickie]

Your approach of considering the rods in two states, aligned with or against the field, is a good starting point. However, you also need to consider the interactions between the charged bead and the rest of the beads in the string.

First, let's consider the case where only the last bead is charged. In this scenario, the charged bead will experience a force in the positive y direction, but the rest of the beads will not be affected by the electric field. This means that the probability of a rod being aligned with or against the field will not change, and the average length of the string will remain the same as in the absence of the electric field.

Now, let's consider the case where all the beads in the string are charged. In this scenario, the charged beads will experience a force in the positive y direction, and they will also exert a force on the adjacent beads. This will lead to a stretching of the string, resulting in an increase in the average length of the string.

To determine the exact value of the average length, you will need to consider the interactions between the charged beads and the rest of the string, and also take into account the temperature of the fluid. This can be done using statistical mechanics and the equation you have provided:

\sum\frac{e^{-\beta E}}{Z}

Where E is the energy of the system, Z is the partition function, and \beta = 1/(kT) where k is the Boltzmann constant and T is the temperature.

Overall, your approach is on the right track, but you will need to consider the interactions between the charged beads and the rest of the string to determine the exact value of the average length.