The magnetic rigidity is the momentum per unit charge of a particle. It is a quantity of great importance in the field of accelerator physics and in the study of cosmic rays.
The Lorentz force, in the absence of an electric field, may be written
q v × B=dp/dt
where q is the particle charge
is its velocity, B
is the magnetic field it is moving through, p
is the particle momentum and d/dt is the derivative with respect to time.
Let us consider the situation in which the particle is travelling at right angles to the magnetic field lines. In that case the left hand side of the above equation becomes the product qvB. The momentum vector will change direction as it moves through the magnetic field. We introduce a differential angle dθ. This allows us to write the differential momentum in the form dp=pdθ and it simply follows dp/dt=pdθ/dt. The angular velocity is equal to the velocity v divided by the radius of curvature of the particle trajectory ρ. So the right hand side of the first equation becomes pv/ρ. Putting everything together, it is evident that the velocity terms cancel and rearrangement yields
This is the magnetic rigidity expressed in terms of the magnetic field normal to which the particle is travelling and its bending radius. In accelerator physics the momentum is often expressed in GeV/c. This allows the formula for the magnetic rigidity to be written in the following simple form
The formula for the magnetic rigidity makes clear the trade off accelerator designers make in deciding the bending magnetic field B and the size of the machine (related to ρ). As an example, consider the LHC at CERN, the highest energy accelerator built to date (currently it is being commissioned). Two 7 TeV proton bunches will be smashed together in the search for the Higgs Boson. The magnetic rigidity of the protons in the LHC will be an enormous 23,350 Tesla meters (calculated using the last equation above). The LHC is built in a pre-existing tunnel with bending radius 2.8 km. The magnetic rigidity relation shows the magnetic field in the bending magnets must be 8.3 Tesla. This is far beyond the capabilities of standard magnets and so superconducting magnets must be used. In order to use cheap conventional magnets with bending field around 2 Tesla, the bending radius of the LHC would need to be increased to over 11 km.