Chapter 3

# Electromagnetic levitation

Take 20 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.

A long, half infinite, vertical solenoid of circular section (of radius and made of joined whorls by length unit) is travelled by a current intensity :

A circular coil made of whorls of radius , of résistance , inductance and mass is placed above the solenoid, distant from to its end.

The position of the coil is marked by angle

Electromagnetic levitation
How can a barbecue be made with magnetic levitation ?

The magnetic field due to the solenoid at the center of the whorl is given by :

With :

## Question

Determine the electric equation verified by the current induced in the coil.

We will write :

.

### Hint

Calculate the mutual induction of the two circuits and use complex notation to solve the electrical equation.

### Solution

Suppose the field is uniform at the surface of the whorl. Its flux through the coil is :

A mutual inductance between the whorl and the solenoid can be defined :

The electric equation of the coil is then :

In a forced sinusoidal state :

So :

Hence :

It can be deduced :

And :

## Question

Show that the radial magnetic field in the whorl can be written :

### Hint

The magnetic field can be obtained by expressing the conservation of the magnetic flux.

### Solution

Let's compute the radial component of the field.

Outside the axis, the magnetic field can be expressed :

We can notice that the plane défined by the point M and is a anti-symetrical plane for the current distribution along the wire, so the magnetic field is in this plane and has no coordinate along .

Outside the axis, the magnetic field can be obtained by expressing the conservation of the magnetic flux :

Electromagnetic levitation

A small cylinder, centered on axis Oz, of height and small radius is used.

The flux, exiting through this closed surface must be equal to zero :

Thus :

So :

## Question

Deduce the Laplace's force applied to the whorl.

Under which conditions is levitation possible ?

Is this possible equilibrium stable or unstable ?

### Hint

Use the definition of the Laplace force and show that only the magnetic fied along is usefull.

### Solution

The Laplace's force (mainly vertical) is :

So :

However :

And :

(Use )

Thus :

The force becomes :

Hence :

The mean value : (using )

With :

It comes :

It is indeed a repulsive force (oriented upwards).

Right above the solenoid, .

Levitation is possible if the repulsion force is superior to the weight.

It is a stable equilibrium : if the mass goes up, the repulsion force diminishes and the mass goes back down.

Same if the mass begins by going down.

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Self-inductance, mutual inductance