

Distortions in a nonuniform magnetic field. 




Last modified: 14:00 Wednesday 01Aug2001 
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.
Let’s 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 aren’t depending from the direction of the E field.
Let’s 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 rewrite (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:
_{} (8’)
_{} (9’)
Now let’s 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:
_{} (13’)
_{} (14’)
So for both of this cases we can calculate the distortions.
_{}
So the master equations for calculating the distortions in a (slightly) nonuniform 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.