FACULTY OF SCIENCE, WINTER 2008
PHYS 126 LEC B3 : Fluids, Fields and Radiation
(Instructor: Marc de Montigny)
Walker, Physics, Chapter 23: Magnetic Flux and Faraday's Law on Induction
- Section 23-1: Induced Electromotive Force
- In Section 22-6 and 22-7, we saw that a electric current generates a magnetic field
(Eq. 22-9, B = μ0I/2πr, Eq. 22-11, B = Nμ0I/(2R),
Eq. 22-12, B = μ0(N/L)I = μ0nI). The main subject of
this chapter is that the converse is also true: a magnetic field can generate an
electric current.
- Applications: the magnetic torque acting on a loop of current (that is,
conversion of electric energy into mechanical energy) is the basic principle of
electric motors. The converse, discussed here, where mechanical energy is converted
into electric energy, allows us to construct electric generators.
- P. 769, Basic features of magnetic induction: An induced current occurs when there
is a change in the flux of magnetic field going through a loop of current. As we will
see in the next three sections, this change is due to three possible causes: (1) a change
of the magnitude of B, (2) a change of the area of the loop, (3) a change of direction of
the loop relative to the field B.
- Section 23-2: Magnetic Flux
- P. 770, Eq. 23-1, Magnetic Flux: Φ = BA cosθ.
- Unit: 1 weber (Wb) = 1 T•m2
- P. 770, Figure 23-3 shows the magnetic flux through a
loop at different angles.
- The flux Φ is analogous to the volume flow rate discussed in Chapter 15.
- P. 797, Problem 6
- Section 23-3: Faraday's Law of Induction and Section 23-4: Lenz's Law
- Often referred to as Faraday-Lenz's Law of Induction.
- P. 772, Eq. 23-4, Faraday's Law of Induction:
ε = - N ΔΦ/Δt
- P. 775, Lenz's Law: An induced current always flows in a direction that opposes
the change that caused it. The minus sign in Eq. 23-4 indicates that the induced
emf opposes the change of magnetic flux.
- Generally, in this course we consider situations where only one factor (B, A or θ)
will change.
- Change of B: |ε| = NAcosθ ΔB/Δt
- Change of area: |ε| = NBcosθ ΔA/Δt
- Change of direction: |ε| = NAB Δcosθ/Δt
- P. 802, Problem 80
- P. 798, Problem 11
- P. 770, Figure 23-2 illustrates Lenz's
Law when B changes. This is shown also in P. 775, Figure 23-8.
- This figure illustrate Lenz's Law when A
changes: a current is induced
in the ring when it is at locations 2 and 4. At location 2, Φ increases so that, in
order to oppose this increase, Iinduced generates B in the opposite
direction. At location 4, Φ is decreasing so that, in
order to oppose this, Iinduced generates B in the same
direction as the original B.
- P. 801, Problem 67
- Section 23-5: Mechanical Work and Electrical Energy
- P. 778, Figure 23-13 shows the basic
concept of Motional emf. It can be understood in two related ways :
- As a magnetic rod is moving in a magnetic field, the associated magnetic
force causes positive charges inside the rod to move downward. This separation
of charges leads in turn to an electric field, up to the point where
FE = FB, in which case E = vB, thus producing
an emf ε = El = vBl (l = length of rod).
- Another explanation is provided by Faraday-Lenz's Law of electromagnetic
induction (see P. 778). As the rod is moving to the right in
Figure 23-13, ΦB increases
because the area is increasing. Therefore Bind
opposes the original B (out). This means that
Bind (in) corresponds to clockwise Iind.
The magnitude of the induced emf is also ε = vBl.
- P. 778, Eq. 23-5, Motional emf: |ε| = Bvl
- P. 778, Eq. 23-6, E = Bv gives the magnitude of the electric field
corresponding to Eq. 23-5.
- P. 799, Problem 29
- Section 23-6: Generators and Motors
- P. 782, Figure 23-14 shows the
operating elements of an electric generator. Its basic principle is
the electromagnetic induction: the angle θ in Eq. 23-1 is
the varying quantity.
- P. 782, Eq. 23-11, Emf produced by an electric generator:
ε = NBAωsin(ωt)
- Proof of Eq. 23-11 (requires calculus):
ε = -N Δ(BA cos(ωt))/Δt =
-N BA Δ(cos(ωt))/Δt =
-N BA (-ω sin(ωt)), because from calculus,
(cos(kt))' = -k sin(kt).
- Eq. 23-11 describes an alternating emf (and current)
which oscillates between εmin = - NBAω
and εmax = + NBAω with an angular
frequency ω, or period T = 2π/ω.
- P. 783, an electric motor is essentially the inverse of an
electric generator: it uses electrical energy to produce mechanical work.
- P. 799, Problem 36
- Section 23-7: Inductance
- From Eq. 23-4, |ε| = N|ΔΦ/Δt|, and the fact that
B (in Φ) is proportional to the source-current I, we find that
|ε| is proportional to |ΔI/Δt|. The constant of
proportionality, L, is called inductance. L may also be seen
as the constant of proportionality between NΔΦ (the total flux)
and ΔI.
- Two types of inductance
- Mutual inductance M: when two coils are considered.
M is the constant between ε produced in one coil by the
current I in the other coil. (We will not consider it any further hereafter.)
- Self-inductance L: only one coil is considered, and L
is the constant between ε and the current through that coil.
- P. 785, Eq. 23-12, Inductance L defined by
|ε| = N|ΔΦ/Δt| = L |ΔI/Δt|.
[SI Unit: 1 henry (H) = 1 V•s/A]
- P. 785, Eq. 23-14, Inductance of a solenoid:
L = μ0 (N2/l)A =
μ0 n2Al
- Remark: for an inductor, V = LΔI/Δt; for a capacitor, V = Q/C
= (integral of I)/C; for a ohmic resitor, V = RI.
- Section 23-8: RL Circuits
- P. 787, Figure 23-19 represents an RL circuit,
that is, a resistor R and an inductor L in series with a battery of emf ε.
- P. 787, Eq. 23-16, Increasing current versus time (after closing the switch):
I = (ε/R) (1 - e-tR/L)
- P. 787, Figure 23-20 describes Eq. 23-16.
- P. 787, Eq. 23-15, Time constant τ = L/R
- P. 800, Problem 46
- Section 23-9: Energy Stored in a Magnetic Field
- Even in a absence of a resistor, a battery connected to an inductor must do
electrical work in opposition to the inductor's self-induced back emf. This energy
is not dissipated, it is stored in the magnetic field, just as the energy in a
capacitor is stored in the electric field (see Eq. 20-19 in
Chapter 20).
- P. 789, Eq. 23-19: U = ½LI2 (Proof in P. 788)
- P. 789, Eq. 23-20, density of energy uB = B2/(2μ0)
(see Proof in P. 789 for the solenoid).
- P. 800, Problem 55
- Section 23-10: Transformers
- Transformes change values of current or voltage from a source to
an electrical device (CD player, TV, lightbulb, etc.).
- P. 790, Figure 23-22 shows the basic elements
of a transformer.
- The primary coil refers to the source, whereas the secondary coil refers
to the electrical device.
- The iron core ensures that the magnetic fluxes are
equal: ΔΦp = ΔΦs. Therefore
the relations εp = - Np ΔΦp/Δt
(primary coil)
and εs = - Ns ΔΦs/Δt
(secondary coil) lead to the following equation.
- P. 791, Eq. 23-22, Transformer Equation for Current and Voltage:
Is/Ip = Vp/Vs =
Np/Ns
- P. 800, Problem 58
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