Magnetism & Electromagnetism
A magnetic field is a region in which a magnetic material experiences a force, and is caused by a moving charge or a permanent magnet.
Just like gravitational and electric fields, magnetic fields are represented by field lines - except instead of talking about positive and negative end, we talk about north and south poles. The closer the field lines, the stronger the magnetic field. Identical to charges, like poles repel and opposite poles attract.
The diagram above shows the field patterns from permanent bar magnets, but magnetic fields arise all the time from current-carrying wires:
For a straight current-carrying wire, the field lines are arranged in concentric circles in a plane perpendicular to the wire and direction of current. This is found using the right-hand grip rule.
A flat current-carrying coil produces the same pattern, but all the way around the coil, so the field is three-dimensional.
A solenoid produces this pattern on an elongated scale.
Magnetic Flux Density
A current that is perpendicular to a uniform magnetic field will induce a force. The direction of this force can be established with Fleming's left-hand rule:
The magnitude of the force causing the motion is proportional to the magnetic flux density, B, of the magnetic field. This is defined as the force on one meter of wire carrying a current of 1 amp, perpendicular to the magnetic field.
F = BIL sinθ Force = flux density x current x length of wire x sin(angle with magnetic field)
If the wire is perpendicular, the equation is just F = BIL, as sin(90°) = 1
If the wire is parallel, there is no force, as sin(0°) = 0
This relationship can be investigated with a known length of wire on a scale, running perpendicular through a uniform magnetic field. When a known current passes through the wire, the scale will read a change in mass. This, multiplied by g, is equivalent to the force exerted on the wire.
The units of magnetic flux density are Tesla, T, defined as 1Wb/m (see electromagnetism, below)
Motion of Charged Particles
F = BIL is the equation used for calculating the force on a current-carrying wire perpendicular to a uniform magnetic field. This is not to be confused with the following equation, used for charged particles perpendicular to a uniform magnetic field:
F = BQv Force = flux density x charge x velocity
According to Fleming's left hand rule, the motion of a charged particle must always be perpendicular to its motion. This means that the articles will travel on a curved or circular path, as the force induced acts as a centripetal force:
A velocity selector is a device that makes use of electric and magnetic fields to produce a fine stream of particles at a given velocity. The two fields are perpendicular to one another, and the particles are charged before entering the device.
The particles will experience an upwards force from the magnetic field, according to Fleming's left hand rule, of magnitude BQv.
The particles experience a downward force from the electric field, because the positive charge of the particle is attracted to the negative end of the field. The magnitude of this force is given as EQ.
Unless the upwards and downwards forces are the same, the particle will experience a resultant force and be deflected.
Therefore, only particles where BQv = EQ will continue in a straight line. The charges cancel, so the selected velocity is given as:
v = E/B selected velocity = electric field strength / flux density
This is used in mass spectrometers to identify ions and particles, based on their mass.
Magnetic flux, ϕ, is defined as the product of the perpendicular component of the flux density through a given area. More or less, this means the number of field lines passing through a particular area. It is calculated as:
ϕ = BA cosθ magnetic flux = flux density x area x cosθ
Magnetic flux linkage, Nϕ, is the product of the flux, ϕ, and the number of coils, N.
ϕ = BAN cosθ magnetic flux linkage = flux density x area x number of coils x cosθ
Both flux and flux linkage have Weber, Wb, as their units. However, to avoid confusion, Weber Turns are sometimes used as the units for flux linkage.
When a conductor moves through a magnetic field and experiences a change in magnetic flux, an e.m.f. is induced. This happens because the electrons in the conductor move to one side of it, according to Fleming's left-hand rule. This leads to a resultant positive charge on the opposite side.
Faraday's Law states that the magnitude of e.m.f. induced is directly proportional to the rate of change of magnetic flux linkage:
ε = -Δ(Nϕ) / Δt e.m.f. = - change in magnetic flux linkage / change in time.
ε = -Δ(BAN cosθ) / Δt (using the equation for flux linkage above)
Lenz's law states that the direction of the induced e.m.f opposes that of the change in flux that has taken place. This is why Faraday's law above has a negative sign in it.