Inductance
From the Faraday law, we find that a changing current in one circuit induces emf in a nearby circuit. This is because, it produces a changing B-field and hence also produce an E-field. This induced E-field creates emf in the second circuit and drives current in it. The emf is obtained by the usual Flux rule,
ε = -dφ/dt. The flux itself is proportional to the current in the first circuit and the proportionality constant is called Mutual Inductance between the two circuits.
ε = -dφ/dt. The flux itself is proportional to the current in the first circuit and the proportionality constant is called Mutual Inductance between the two circuits.
The Demo of FM radio shown in class was to show how changing current in one coil creates current in the second coil and the magnitude of this induced current (and hence the flux) depends on the geometrical placing of the two coils.
The mutual inductance between the two coils is
M= ∫∫ (μ0/4 π) dl1.dl2/|(r1-r2)|, though this is seldom used in calculations.
M= ∫∫ (μ0/4 π) dl1.dl2/|(r1-r2)|, though this is seldom used in calculations.
A changing current in a circuit also produces an emf in this circuit itself for the same reason of creation of ionduced E-field. . The flux linked to the circuit is proportional to the current itself and the proportionality constant is called self inductance L. The back emf is ε = -L di/dt.
When a current passes in a circuit, there is an E-field in the wire. The current density J is proportional the the NET E-field existing in the wire at that point, . This net E-field may have contribution from any accumulated charge on the surface of the wire or induced E-field or any other source. J = σ E where sigma is the electrical conductivity.
Ampere-Maxwell Law
Just as changing magnetic field is accompanied by E-field, changing E-field is accompanied by B-field. Any that creates time varying E-field will also produce B- field. Thus curl of E is related to ∂ B/ ∂ t for the E field corresponding to B-field. Curl E is also related to J. Doing dimensional analysis and using charge current continuity equation, one gets
∇ X B = μ 0 J +ε 0 μ 0 ∂ E / ∂ t.
This is known as Ampere-Maxwell law and it was Maxwell who worked out this extra term (from a different analysis)
Poynting Vector
The magnetic field energy is B2/2 μ 0 per unit volume. Electrical energy is ε0E2/2 per unit volume. The Poynting vector S is defined as S =(E X B)/μ0. The direction of S gives the direction in which EM energy travels and magnitude of S gives energy flowing per unit time per unit area (perpendicular to the direction of flow)
For any volume, ∫ S.da =-d/dt(Uem)-∫ (E.J)
∫ (E.J) gives the loss of EM energy due to work done by the fields on the charges as they move to form current. This appears as the thermal energy if the current is in a conducting wire. You can write
∫ S.da + ∫ E.J = -d/dt (Uem)
∫ S.da + ∫ E.J = -d/dt (Uem)
LHS first term gives EM energy going out of the volume through the surface, LHS second term gives energy of EM fields spent on charges (which appears as thermal energy but we are accounting only EM energy and not the total energy). The it gives loss of EM energy in the volume per unit time which is same as the RHS.
