3电磁感应

ε

0
NBSω Peak value of emf
Direction of induced EMF:
Positive directions for and : Right hand rule Find out n direction by choosing loop direction.
motional
(v B) dl
(2)

d dt
where : B dS
S
If not closed，add an imaginary curve to make it closed or get d/dt by the area swept per unit time
F B C
F When at equilibrium, E v B e D F D D dl (v B) dl vBdl vBl C e C C
∵ vl is the rate at which the area changes ∴ vBl is the rate at which the flux changes, = d/dt EMF only exist on the moving part CD, not on the other part.
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Discussion:
1. Metal conductor, R small ( small), If big. 2. If ∝ ω,— High frequency oven
High frequency AC power
3. If ∝ R-1 , —— Ferrite, stalloy ； 4. If opposite to I, —— Eddy drag, asynchronous motor
d d N dt dt
Exmp 1：
l = 10cm, B = 1000Gs, v = 1.0m/s, look for Solution: Flux of the loop:
×
×
× × ×
l × B×
×
v
dΦ dx Bl Blv ε ( B) l dt dt 0.10T 0.10m 1.0m/s 0.01V.
General formula for Motional emf ：
motional
(v B) dl
EMF can be generated on the moving conductor, but the current may not be able to be generated. EMF can be generated on the moving conductor only when the conductor cuts the magnetic field lines.
v
Example
A straight wire of length L is rotating at angular velocity in a uniform magnetic field B. Find emf BA and potential difference UBA . Sol.：(1) take dl at a distance l from B v B same direction with dl , v = l
Φ Blx
x
×
Exmp 2: Rotating loop in uniform magentic field: Solution: Φ BS cos θ S θ B dΦ dθ ε N NBS sin θ NBS sin t dt dt ε0 sin ωt where
L
ε ε
B
L
ε
B
(c) Φ < 0 ，dΦ < 0 > 0, along with L
( d ) < 0 ， d > 0 < 0, opposite to L
Discussion: 1. For any loop L, is opposite to 2. Magnitude of only depends on
2.1 Motional EMF
Magnetic force on electrons in conductor CD F = - e(v B) (downward) I counterclockwise direction (caused by emf on CD) emf = work on unit charge by Lorentz force as it moves from C to D A I D v
For a loop composed of multiple turns (solenoid), the induced emf: d 1 2 N d N d1 d 2 d dt dt dt dt dt is called fluxoid or flux linkage.
1 dΦ If R dt
B
dΦ dB S dt dt
We choose
~
B μ0 nI
I I 0 sin ωt
Eddy current
then
μ0 nSI 0 ω If cos ωt R
An Induction Stove
The water in the metal pot is boiling. Yet, the water in the glass pot is not boiling, and the stove top is cool to the touch. The stove operates in this way by using electromagnetic induction.
Chapter III
Electromagnetic Induction
§1 Law of Electromagnetic Induction
1.1 Electromagnetic induction phenomena
battery
switch
A current can be induced in a loop as the magnetic flux through the loop changes with time. Induced electromotive force (emf) can be induced in a loop as the magnetic flux through the loop changes.
,L , n
B
>0 when is acute angle; <0 when is obtuse angle. Steps to find out the direction of EMF: •Direction of L •Direction of n
•Sign of
•Sign of d/dt •Direction of
（ q total charges）
Lenz’s Law
Heinrich Friedrich Emil Lenz 1804 – 1865 Russian physicist
1.3 Lenz’s Law The magnetic flux produced by the induced current always tends to oppose the flux change that causes the induced current (or the induced emf).
fA pointing to left is a result of energy conservation law.
1.4 Eddy current
Varying magnetic field can induce currents entirely within a conducting material. Eddy current
§2 Motional EMF and Induced EMF

d dt
S
where : B dS
Three ways to change the flux ：
– change the area S ( move the loop ) —— motional EMF – change the field B —— induced EMF – change both B and S
1 BA B (v B) dl 0 Bldl BL2 2 ( 0, direction：B U AB BA
1 BL2 2
B： low potential，negative charges A： high potential，positive charges
along with L when > 0; opposite to L when < 0.
n
L
B
L
n
B
ε
n
ε
n
( a ) Φ > 0 ， dΦ > 0 ε < 0 ， ε along with L
( b ) Φ > 0 ， dΦ < 0 ε > 0, opposite to L
q Idt
t1 t2
dΦ dt dΦ , dt
Φ2
not on
1 dΦ I R dt
1 1 q dΦ Φ2 Φ1 R Φ1 R
3.
dΦ —— dt
rate of change of B determines
Φ Φ2 Φ1 —— changes of flux B

d dt
or

d -k dt
Induced emf = k time rate of change of flux in SI unit（V, Wb, s） k = 1.

d dt
If flux of every turn is the same , then =N.
In the opposite direction for > 0
In the same direction for < 0
Magnetic force on induced current always tends to oppose the motion of the conductor. I ∵ v towards right ∴ fA left（ fA= Idl B ） ∴ I counterclockwise ( or： > 0 ∴ flux of I in opposite direction ) Work and Energy： Work by external force I 2Rt ( heat energy ) （constant velocity，kinetic energy not changed） R fA v
v o

B in the formula does not change with time
Calculation of Motional EMF
Two ways to calculate Motional emf ：
(1)
where v、B might not be uniform，so could not be brought out in front of the integral sign
electromagnetic induction.
1.2 Faraday’s Law of induction
The induced emf around a closed mathematical path in magnetic field is equal to the rate of change of the magnetic flux intercepted by the area within the path: