# A question from a Physics 2 exam: About electron cloud an a constant external field:

I am trying to prepare for my exam in Physics $2$, the following is a question from an old exam (the question also have a detailed answer to it, but I don't really understand it).

An acceptable model for an atom contains a point charge $+ze$ (nucleus) at the center of a ball charged of charge $-ze$ with uniform density $\rho$. The radius of the ball is $a$.

An atom is placed under a uniform field $\vec{E}=E\hat{x}$, as a result, there are forces on the nucleus and the electron cloud at opposite directions.

1. Assuming that the electron cloud moves, but not distorted: calculate the relative shift $\delta$ between the nucleus and the center of the electron cloud (assume $\delta<a$)

2. Assuming that the electrical field was increased gradually from $0$ to its final value $\vec{E}$, calculate the work, $W$, the electrical field did on the atom.

There are another parts to the question that I omit for now, since they are not relevant to my question.

Note: I have translated the question, so the English problems origin is me, if something is not clear please ask and I'll try to answer.

The answer to the first question finds $\delta$ s.t that the electric force being applied from the electron cloud equals (and is opposite in direction) from the electric force being applied from the electric field $\vec{E}$.

The answer to the second question say that

the field is increased gradually so the atom is in equilibrium at each step of the process, the force the electrical field need to overcome is the attraction force between the nucleus and the electric cloud. For a given distance from the center $x$, this field is given by $E(x)=\frac{ze}{a^{3}}x$

The work is then calculated by $\int_{0}^{\delta}F(x)\, dx$ where $F=qE$.

Note that if I calculated correctly, the calculation $E(x)$ corresponds to the field of a ball with uniform charge density $\rho$ and with total charge $-ze$.

My questions are:

1) In the solution to the first question, the force of the attraction is calculated as if there was no external field $\vec{E}$.

Shouldn't the electron cloud move somehow so that this calculation was not good anymore ? why can we still consider the ball as a ball with uniform charge density $\rho$ ?

2) In the solution to the second question, what does it mean that the atom is in equilibrium at each step of the process ? if this is the case the nucleus shouldn't be moving at all.

Also, if I change the field gradually, why does the atom is in equilibrium state ? I thought that for small values of $\vec{E}$ total electrical force is $-\hat{x}$, since the force of the attraction is higher than the force from the external field, and so the nucleus would go back to the center and then pushed to the right and goes back to the center again, until $|\vec{E}|$ reaches some minimal value to overcome the attraction.

"why can we still consider the ball as a ball with uniform charge density ρ ?"

In reality, we cannot. The question doesn't ask about a realistic model. The electron cloud is considered to remain spherical with constant density simply by hypothesis. This is what I infer from "Assuming that the electron cloud moves, but not distorted".

"what does it mean that the atom is in equilibrium at each step of the process ? if this is the case the nucleus shouldn't be moving at all."

Yes, a true equilibrium process is not possible for this reason. But imagine the process takes 1 microsecond. Then imagine it takes one second. Then imagine it takes one year. As we allow the process to take longer and longer times, it can get closer and closer to being in equilibrium at all points. The work done on the atom will always be slightly higher than that calculated in the problem, but it can be made arbitrarily close by taking long enough times. The question is asking about the limit.

"Also, if I change the field gradually, why does the atom is in equilibrium state ? I thought that for small values of E⃗ total electrical force is −x^, since the force of the attraction is higher than the force from the external field, and so the nucleus would go back to the center and then pushed to the right and goes back to the center again, until |E⃗ | reaches some minimal value to overcome the attraction."

I find your wording very confusing, but I think you are asking why there are no oscillations. That is again by assumption - it is assumed the atom is always in equilibrium. If it were oscillating, it would have had to have been out of equilibrium at some point. Again, this is only an idealized model.