Chapter 11

# Reflection of a wave on a "perfect" metal, radiation pressure (wave calculation)

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A MPPW, linearly polarized, propagates in the vacuum in the (Ox) axis, in the direction of increasing (assuming ) :

At , it arrives on the plane surface of a metal mirror perfectly conducting, in which the fields and are zero, and gives birth to a reflected wave propagating in the direction of decreasing :

## Question

By writing the boundary conditions that must check the fields and in , determine :

• The amplitude of the field reflected according to .

• The surface charge and the surface current that can be found on the metal surface at .

### Hint

What are the boundary conditions of an electromagnetic wave passing through a perfect metal ?

### Solution

The tangential component of the electric field should be continuous, therefore :

The boundary condition for the electric field :

Show that .

The boundary condition for the magnetic field gives :

The incident magnetic field is :

The reflected magnetic field is :

The resulting field in the vacuum for is :

We can deduce :

Real notation :

## Question

Determine the resulting electromagnetic field of the incident wave and the reflected wave in the half-space .

Briefly characterize the resulting wave.

Calculate the average value of the Poynting vector.

### Hint

• The total field is the sum of the initial field and reflected.

• What is a standing wave ?

### Solution

The resultant electric field is :

Either in real notation :

Similarly, for the magnetic field :

Either in real notation :

This type of solutions, called standing plane wave is very different from a progressive plane wave. Spatial and temporal dependencies occur separately  : spatial dependence occurs in the amplitude of the time oscillation and no longer in phase, so that all points vibrate in phase or in opposition of phase.

At certain points (called nodes of vibrations), the field is always zero.

In other points (called vibration antinodes), the amplitude of vibration is maximum.

We can calculate Poynting vector and verify that its average value is zero : a standing wave does not transport energy.

## Question

The electromagnetic field exerted on a mirror surface a force whose expression is, in real notation :

Offer an explanation for the presence of the factor .

Derive that the wave exerts a pressure on the mirror whose average value will be calculated as the function of the average energy density of the incident wave ( is called radiation pressure).

### Solution

The electromagnetic field exerted on a mirror surface  a force whose expression is, in a real notation : (note that )

Indeed, the surface is subjected to the action of the magnetic field external to it ; so do not take into account the magnetic field created by the charged surface and traveled by volume current.

The factor takes into account the remarks.

The average value of the force becomes :

The average radiation pressure is then :

The volumetric energy density of the incident wave is :

Whose average value is :

We can deduce the radiation pressure :

We can notice that :

Where is the density of photons of the incident wave and the energy of one photon.

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