Chapter 1

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 .

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Electrostatics
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Charged ring equivalent to a dipole