# Homework Help: Van der waals attraction between 1-D wires

1. Nov 16, 2014

### CAF123

1. The problem statement, all variables and given/known data
Consider two infinitesimally thin (i.e 1-D) wires of equal length $L$, and at
mutual distance $d$.
Consider the two configurations shown in attachment
Estimate the van der Waals interaction between the wires, for $d \gg L$, in the two situations

where: (i) the wires are parallel and in register, as on left and (ii) the wires lie on parallel
planes and are in the ''cross'' configuration as on the right.

2. Relevant equations
VDW attraction between atoms modeled by $u(r) = - C/r^6, C$ a constant.

3. The attempt at a solution
Set up a coordinate system with $z$ axis coinciding with the wire with origin midway. The attraction between the atom at $z=0$ on one wire and an arbritary atom at some distance $\sqrt{d^2 + z^2}$ is therefore $u(z) = -C/(z^2 + d^2)^3$. This atom at $z=0$ therefore contributes $U(r) = \int_{-L/2}^{L/2} u(z) dz$, i.e the interaction energy contributed from this atom on one wire and all others on the other. Now how should I proceed to get the total interaction energy due to all atoms? I wasn't specified the atom density along the wires.

I can't make sense of the mutual distance of the wires being a distance $d$ apart in the second configuration.

Thanks.

#### Attached Files:

• ###### VDW.png
File size:
174 bytes
Views:
59
2. Nov 17, 2014

### CAF123

The question was updated with the atomic density being $\sigma$, so my answer for the total interaction energy between the wires is $\sigma L U(r)$ Is it correct?

3. Nov 18, 2014

### CAF123

Can anyone help?