# Electrostatics and Magnetostatics

# Treatment of water

Take 15 minutes to prepare this exercise.

Then, if you lack ideas to begin, look at the given clue and start searching for the solution.

A detailed solution is then proposed to you.

If you have more questions, feel free to ask them on the forum.

The half metallic space is under a uniform potential .

The half-space
is filled with an electrolyte made of cations K^{+} and anions A^{-}_{}.

The problem is translation invariant along and axis.

The potential only depends on .

At thermodynamic equilibrium, under temperature , the density of ions (anions and cations) in the electrolyte is expressed with the Boltzmann distribution :

and

## Question

Quote another situation where the Boltzmann distribution is used.

Explain the physical interpretation of this law.

### Hint

Think about potential energy of a punctual charge.

### Solution

These expressions involve the potential energy of an ion of charge in the electric potential V in the term , called the Boltzmann distribution.

A similar result can be found in kinetic theory of ideal gas, when studying the isothermal atmosphere.

## Question

Express the density of charge as a function of , , , and .

As long as , deduce that the potential is the solution to an equation of this kind :

Where is a constant that you should express as a function of , , , and .

### Hint

Think about Poisson's equation.

### Solution

The total charge density is :

Poisson's equation, leads to :

With , the equation becomes :

With :

## Question

Determine as a function of , and and considering that .

Deduce the expression of the electric field in the electrolyte as a function of , and .

Why can we talk about electric field screening ?

### Hint

### Solution

Considering the interface conditions, the solution to this equation is :

The field is given by :

In the absence of electrolyte, the electrostatic field would be uniform.

The electrolyte acts like a screen to the field on a characteristic distance .