Quadrupole magnets consist of a group of four magnets laid out so that in the planar multipole expansion of the field, the dipole terms cancel and where the lowest significant terms in the field equations are quadrupole. Quadrupole magnets are useful as they create a magnetic field whose magnitude grows rapidly with the radial distance from its longitudinal axis. This is used in particle beam focusing. The simplest magnetic quadrupole is two identical bar magnets parallel to each other such that the north pole of one is next to the south of the other and vice versa. Such a configuration will have no dipole moment, and its field will decrease at large distances faster than that of a dipole. A stronger version with very little external field involves using a k=3 Halbach cylinder. In some designs of quadrupoles using electromagnets, there are four steel pole tips: two opposing magnetic north poles and two opposing magnetic south poles. The steel is magnetized by an electric current in the coils of tubing wrapped around the poles. Another design is a Helmholtz coil layout but with the current in one of the coils reversed.
Quadrupoles in particle accelerators
At the particle speeds reached in high energy particle accelerators, the magnetic force term is larger than the electric term in the Lorentz force:
F = q ( E + v × B ) , {\displaystyle \mathbf {F} =q(\mathbf {E} +\mathbf {v} \times \mathbf {B} ),}
and thus magnetic deflection is more effective than electrostatic deflection. Therefore a 'lattice' of electromagnets is used to bend, steer and focus a charged particle beam The work done on a relativistic particle (=the energy needed) in order to keep it in a circular motion with a give radius is equal, regardless of the source of the centripetal force. There is nothing more efficient or effective in using B or E fields in creating circular motion, given setup corresponding to the properties of that field. There are 2 main reasons why particle accelarators are using B fields instead of E fields for radial deflection: 1. B fields always a force perpendicular to the direction of motion and therefore it is much easier to build a setup for circular motion using this type of field 2. In relativistic energies the Energy required to create this motion is so large, that when an E field is used, it causes vacuum-electrical breakdown of the materials used in the system. This doesn't happen with a B field precisely because the Lorentz force induced by it is proportional to the velocity - if a particle doesn't move fast enough, it won't experience a strong enough force to break down the material it is in, even though the same amount of energy would be used in order to produce the motion. Calculation: The force required for circular motion with radius r:
F r = γ m v 2 r = p v r {\displaystyle F_{r}={\frac {\gamma mv^{2}}{r}}={\frac {pv}{r}}}
Compare with the Coulomb force or the electric component of the Lorentz force:
p v r = q E ⇒ r = p v q E {\displaystyle {pv \over r}=qE\Rightarrow r={pv \over qE}}
Magnetic component:
p v r = q v B ⇒ r = p q B {\displaystyle {pv \over r}=qvB\Rightarrow r={p \over qB}}
r = r:
p q B = p v q E ⇒ E = v B ∼ c B {\displaystyle {p \over qB}={pv \over qE}\Rightarrow E=vB\sim cB} (for relativistic particles) Energy density for each field:
u E = 0.5 × ϵ 0 E 2 {\displaystyle u_{E}=0.5\times \epsilon _{0}E^{2}}
u B = 0.5 B 2 / ( μ 0 ) {\displaystyle u_{B}=0.5B^{2}/(\mu _{0})}
Divide by each other and substitute E = cB:
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