Lecture 21 – Faraday’s Law & Inductance
Chapters 35 & 36 - Tuesday April 3rd
•Review of Faraday’s law
•Examples using Faraday’s law
•Definition of inductance
•LR circuits
•Energy stored in inductors and magnetic fields
•Oscillations - LC circuits
Reading: pages 823 thru 835 (all of Ch. 35) in HRK
Read and understand the sample problems
WebAssign deadline (chs 34 and 35) tomorrow at 11:59pm
Ch. 36: to be posted on-line later today or tomorrow
Practice problems from Ch. 36: E11, E19, E43, E51
Exam 3 is Tuesday April 17th (in class) – Chs. 33 to 38
Review of motional emf
Magnetic flux:
ΦB = BA = BDx
d ΦB
d
ε =
= (BDx )
dt
dt
BDv
ε
i= =
R
R
B 2D 2v 2
Mechanical Power = F1v = (iDB ) v =
R
2
2 2 2
⎛
⎞
BDv
B
Dv
2
⎟
Electrical Power = i R = ⎜⎜
R=
⎟
⎟
⎝ R ⎠
R
Faraday’s and Lenz’ laws
Magnetic flux:
ΦB =
∫
G G
B ⋅ dA = BA cos θ
1 weber = 1 tesla · meter2
Faraday’s law:
d ΦB
ε=−
dt
Induced emf drives a
current which opposes
the change in the applied
magnetic field. This
required from energy
conservation.
Emf and electrostatic potential
ε=
∫v
G G
d ΦB
E ⋅ ds = −
dt
•In electrostatics, we noted that electrostatic forces are
conservative, because
G G
∫v E ⋅ d s = 0.
•This is no longer true when dΦΒ/dt is finite.
•Consequently, induced electrostatic (quasi-static) forces
are not conservative.
Finally:
∫v
G G
E ⋅ ds =
⇒
G
G
d
∇ × E ⋅ dA = −
dt
G
G
dB
∇×E = −
dt
∫(
)
∫
G G
B ⋅ dA
Induced electric field
ε=
∫v
G G
d ΦB
E ⋅ ds = −
dt
•Looks just like Ampère’s law for magnetic field:
∫v
G G
B ⋅ d s = μ0i
•Therefore, we can use precisely the
same tricks to find the induced
electric field
∫v
G G
E ⋅ d s = E (2πr )
i is really the flux of the current density j. In Faraday’s law, -dΦΒ/dt plays the
same role as μ0i does in Ampère’s law. Note:
ΦB =
∫
G G
B ⋅ dA;
i=
∫
G G
j ⋅ dA
Inductors: the analogy with capacitors
+q
+q
−q
−q
Inductors: the analogy with capacitors
•The transfer of charge from one terminal of the capacitor
to the other creates the electric field.
•Where there is a field, there must be a potential
gradient, i.e. there has to be a potential difference
between the terminals.
•This leads to the definition of capacitance C:
q = C ΔV
•q represents the magnitude of the excess charge on either
plate. Another way of thinking of it is the charge that
was transferred between the plates.
SI unit of capacitance:
1 farad (F) = 1 coulomb/volt
(after Michael Faraday)
Capacitances more often have units of picofarad (pF) and microfarad (μF)
Inductors: the analogy with capacitors
•An increasing current creates magnetic flux.
•As this increasing magnetic flux threads the circuit, an
emf is necessarily generated (Faraday’s law), i.e.
d ΦB
ε =−
dt
Inductors: the analogy with capacitors
•In circuits, we are more concerned with currents and
voltages (including emf’s) than flux.
•However, ΦB is obviously proportional to the current i in
the inductor. Thus, we can assume that
di
ε∝
dt
Inductors: the analogy with capacitors
di
dt
εL
•Therefore, in analogy with the capacitor, a changing
current in a circuit (di/dt) serves as a source of emf (ε),
just like a build up of charge (±q) leads to potential
differences (V) in circuits
•We can, therefore, define a quantity L called inductance,
which relates di/dt and ε
di
εL = −L
dt
Calculating Inductance
εL = −
d (N ΦB ,1 )
dt
=−
d (ΦB ,N )
dt
⇒ Li = N ΦB ,1 = ΦB ,N
Or
L=
N ΦB ,1
i
=
ΦB ,N
i
di
= −L
dt
Energy and Inductance
•Just like the capacitor, the energy is really stored in
electromagnetic fields.
•In chapter 30, we considered the capacitor to be like a
spring. An inductor is like a fly-wheel, storing energy in
the form of the magnetic field produced by a current, i.e.
this energy is a bit like kinetic energy, whereas the
energy stored in a capacitor is more like potential energy.
UB
UB =
∫ dU
0
t
B
=
t
∫ (ε i )dt = ∫
L
0
0
⎛ di ⎞⎟
⎜⎝⎜L dt ⎠⎟⎟ i dt =
i
∫
0
1 2
(Li )di = Li
2
Energy and Inductance
•Just like the capacitor, the energy is really stored in
electromagnetic fields.
•In chapter 30, we considered the capacitor to be like a
spring. An inductor is like a fly-wheel, storing energy in
the form of the magnetic field produced by a current, i.e.
this energy is a bit like kinetic energy, whereas the
energy stored in a capacitor is more like potential energy.
2
UB
Li
uB =
=
=
Al
Al
1
2
1
2
(μ0n 2Al ) i 2
Al
1 2
μ0n 2i 2
B
=
=
2
2μ0
Electromagnetic oscillations
q(t ) = qm cos (ωt + φ)
i(t ) = −ωqm sin (ωt + φ )
1 2 1 q2
U = U B + U E = Li +
2
2C
Electromagnetic oscillations
q(t ) = qm cos (ωt + φ)
i(t ) = −ωqm sin (ωt + φ )
1 2 1 q2
U = U B + U E = Li +
2
2C
ω=
1
LC
Electromagnetic oscillations
Damped (RLC) oscillations
2⎤
⎡
dU
d 1 2 1q
di
q dq
2
⎥ = Li +
= −i R = ⎢ Li +
dt
dt ⎢⎣ 2
2 C ⎥⎦
dt C dt
2
⎛dq ⎞⎟
di
q dq
dq di
q dq
= R ⎜⎜ ⎟⎟ + L
+
=0
i R + Li +
⎝ dt ⎠
dt C dt
dt dt C dt
2
d 2q
dq
q
L 2 +R
+ =0
dt
dt C
q(t ) = qme −Rt / 2L cos (ω't + φ)
2
ω' = ω − (R / 2L )
2