One way to measure the charge to mass ratio of
particles is to send them through a region of space in
which there are both electric and magnetic fields. 
The fields must be (as near as possible) uniform. 
The fields are oriented such that the force due to
the electric field, F_{E} is in the same
direction but opposite sense to the force due to the
magnetic field, F_{M} 

This means that the two fields
must be at 90° to each other as shown here. 
If the particles pass though
the fields undeflected, then the two forces
must have equal magnitudes. 
The diagram above assumes that
the charge, q is a positive charge. 

The force due to the
electric field is independent of the
velocity of the particles. 
The force due to the
magnetic field depends on the velocity of
the particles. 
Therefore, for a given pair of field strengths, E
and B, the
two forces can only be equal for one velocity. 

We can know the speed of the particles if we know the
potential difference, V through which they are
accelerated. 
During acceleration, electrical potential energy is
converted to kinetic energy. 
So we can write 

therefore 

in which we see the charge to
mass ratio, q/m of the particles. 

The force due to the magnetic
field is given by 

where v is the velocity and B is
the magnetic flux density. 

The force due to the electric field is given by 

where E is the electric field strength. 

If these two forces have equal magnitude then the
velocity of the particles is given by 

Therefore, if we find that the particles pass
undeflected though the fields 

and so the charge to mass ratio can be found using 

in which E is electric field strength,
B is magnetic flux density and V is the
potential difference through which the particles were
accelerated to send them into the fields. 

See
also
Measuring the Charge to Mass Ratio of Electrons 