Betatron injected electrons A betatron is a cyclotron developed by Donald Kerst at the University of Illinois in 1940 to accelerate electrons. The betatron is essentially a transformer with a torus-shaped vacuum tube as its secondary coil. An alternating current in the primary coils accelerates electrons in the vacuum around a circular path. injected electrons The equation of motion that governs the transverse direction is If the orbit is stationary: Condition for the equilibrium orbit

Faraday’s disc The homopolar generator was developed first by Michael Faraday during his experiments in 1831. It is frequently called the Faraday disc in his honor. It was the beginning of modern dynamos — that is, electrical generators which operate using a magnetic field. It was very inefficient and was not used as a practical power source, but it showed the possibility of generating electric power using magnetism, and led the way for commutated direct current dynamos and then alternating current alternators. insulating disc with a conducting ring around its circumference conducting axle OR conducting wire OP embedded in the disc the brushes at Q and R complete the circuit OPQR B is uniform and perpendicular to the plane of the disc, directed out of the page The flux is cut as the conducting wire OP sweeps around through an angle f=POQ:

Inductance self-inductance (inductance) InductanceInductance is the property in an electrical circuit where a change in the electric current through that circuit induces an emf that opposes the change in current. In electrical circuits, any electric current, I, produces a magnetic field and hence generates a total magnetic flux, Φ, acting on the circuit. This magnetic flux, due to Lenz's law, tends to act to oppose changes in the flux by generating a voltage (a back emf) in the circuit that counters or tends to reduce the rate of change in the current. The ratio of the magnetic flux to the current is called the self-inductance, which is usually simply referred to as the inductance of the circuit. The term 'inductance' was coined by Oliver Heaviside in 1886. For a given electric circuit, the magnetic field produced at any point is proportional to the current flowing in the circuit. Accordingly, the magnetic flux linking any closed path is also proportional to I. In a network with a varying current, an inductor acts as an emf! self-inductance (inductance)

a) Switch K1 closed, I(t=0)=0 b) Switch K2 closed, I(t=0)=I0

Inductance of a long solenoid Inside the solenoid, the magnetic field is: The number of turns per unit length is n

Mutual inductance mutual inductance Neumann formula What if consider more than one circuit? mutual inductance Neumann formula