The work function and mutual forces between particles

• dRic2
In summary, the conversation discusses the application of the chain rule in finding the force acting on a particle in a gas of electrons subject to Coulomb potential. The equation for the potential energy is given and it is mentioned that differentiation with respect to xi will yield units of force. The approach for solving the problem is also mentioned.
dRic2
Gold Member
Homework Statement
Let us assume that the work function of an assembly of free particles, subject to mutual forces between these particles, depends only on the relative coordinates
$$\xi_{ik} = x_i - x_k$$
$$\eta_{ik} = y_i - y_k$$
$$\zeta_{ik} = z_i - z_k$$
of any given particle ##P_i## and ##P_k##:
$$U = U(\xi_{ik}, \eta_{ik}, \zeta_{ik})$$
Let the coordinate ##x_i## be varied by ##\delta x_i##, thus obtaining the x-component of the force acting at ##P_i##. Show that the quantity:
$$X_{ik} = \frac {\partial U}{\partial \xi_{ik}}$$
can be interpreted as the x-component of the force on ##P_i## due to ##P_k##.
Relevant Equations
.
I tried to apply the chain rule

$$X_{ik} = \frac {\partial U}{\partial \xi_{ik}} = \frac {\partial U}{\partial x_{i}} \frac {\partial x_i}{\partial \xi_{ik}} = \frac {\partial U}{\partial x_{i}}$$

and I got the force x-component of the force acting on ##P_i## I guess.

but I do not know what to conclude from this...

For definiteness consider a gas of electrons subject (obviously) to Coulomb potential. For N electrons you can show that:$$U_E=\frac {1}{8\pi \epsilon_0}\sum_{i=1}^{N} q_i \sum_{j=1}^{ ,N(i\neq j )}\frac{q_j}{\sqrt{\xi_{ij}^2 + \eta_{ij}^2+ \zeta_{ij}^2 }}$$
Will differentiation w.r.t xi yield units of force? I leave it to you to prove the above equation.

Thanks for the reply, but I solved it by considering the frame of reference moving with, for example, the particle ##P_k##. It this manner ##\delta \xi_{ij} = \delta x_i## (because ##\delta x_j = 0##) and so it is obvious that the only work done after the displacement ##\delta x_i## comes from the force on ##P_i## due to ##P_k## (because all the other displacements vanish and thus do not contribute).

1. What is the work function of a particle?

The work function of a particle is the minimum amount of energy required to remove an electron from the surface of that particle. It is a measure of the strength of the attractive forces between the particle and the electron.

2. How is the work function measured?

The work function can be measured experimentally by using a technique called photoelectron spectroscopy. This involves shining light of varying frequencies onto the particle and measuring the energy of the electrons that are emitted.

3. What factors affect the work function of a particle?

The work function of a particle is affected by several factors, including the type of material the particle is made of, its size and shape, and the presence of impurities on its surface. Temperature and electric fields can also affect the work function.

4. What are mutual forces between particles?

Mutual forces between particles refer to the attractive or repulsive forces that exist between two particles. These forces are a result of the interaction between the electric charges of the particles and can be either electrostatic (Coulomb) forces or van der Waals forces.

5. How do mutual forces between particles impact their behavior?

The mutual forces between particles play a crucial role in determining their behavior and interactions with each other. These forces can affect the stability of a particle, its ability to form bonds with other particles, and its movement in a given environment.

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