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Distortions in a non-uniform magnetic field.

 

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Last modified: 14:00 Wednesday 01-Aug-2001

 

The motion of charged particles under the influence of electric and magnetic fields, E and B, may be understood in terms of an equation of motion:

 

†††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† (1)

 

where m and e are the mass and electric charge of the particle, u is its velocity vector, and K describes a frictional force proportional to u, which is caused by the interaction of the particle with gas.

The ratio m/K has the dimension of characteristic time, and we define

††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† †††††††††††(2)

If t >> t the particle is moving with out any acceleration it means that .

Using (2) the drift velocity vector can be determined by the linear equation

 

††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† †††††††††††††(3)

 

We can define etc.

A different way of writing (1) is the following:

 

††††††††††††††††††††††††††††††††††††††††††† (4)

 

where and , denote the unit vectors in the directions of the fields.

And we should never forget that change sign depending from the charge of the particle.

 

We have special interest in the case where E nearly parallel to B.

Letís just remind our self the definition of Cross Product.

Vector A is a cross product of vectors B and C where

and where

;;;

In the case of E nearly parallel to B, we can say that , where .

Using (4) we can find .

 

 

†††††

 

 

 

Mainly we are interested in relations between and .

 

†††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† (5)

 

††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† (6)

 

We can see that these relations arenít depending from the direction of the E field.

Letís make some approximation till the second order of approximation.

In that case:

 

†† ††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† (7)

 

Using (7) the different way of writing (5) is the following:

 

†††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††

 

So using those approximation we can re-write (5) and (6) in the following way:

 

††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††(8)†††††

 

††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† (9)††††

 

As we see these relations depend from the direction of the B field.

And the where the charge of the particle can change the sign of the .

We can easily rewrite those equations in the cylindrical coordinates:

 

††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† ()

††††††††† ††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††()

 

 

Now letís see these equations in the case where, where .

Using (4) we can find three components of the velocity as we have done previously.

 

†††††††††††††††††††††††† ††††††††††††††††††††††††††††††††††††††††††††(10)

†††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† (11)

†††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† (12)

 

Making the same approximation as in the previous case we have:

 

†††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† (13)

 

†††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† (14)

 

The same equations in cylindrical coordinates are:

 

†††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† (13í)

 

†††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† †††††††††††††(14í)

 

 

So for both of this cases we can calculate the distortions.

 

 

So the master equations for calculating the distortions in a (slightly) non-uniform magnetic field are:

 

††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† (15)

 

†††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† (16)†††††††

 

 

 

This image illustrates the distortion in TPC for non uniform magnetic field.

Where .

 

So these equations are the real equations that we use to calculate the distortions in the TPC.