Physics 2102
Jonathan Dowling
Physics 2102
Lecture 18
Ch30:
Inductors & Inductance II
Nikolai Tesla
Faraday’s Law
• A time varying magnetic
FLUX creates an induced
EMF
• Definition of magnetic flux
is similar to definition of
electric flux
r r
! B = # B " dA
S
d! B
EMF = "
dt
B
n
dA
• Take note of the MINUS sign!!
• The induced EMF acts in such a
way that it OPPOSES the
change in magnetic flux
(“Lenz’s Law”).
Another formulation of
Faraday’s Law
• We saw that a time varying
B
n
magnetic FLUX creates an
induced EMF in a wire,
exhibited as a current.
• Recall that a current flows in a
conductor because of the
dA
forces on charges produced by
an electric field.
• Hence, a time varying
B
magnetic flux must induce an
ELECTRIC FIELD!
• But the electric field line
C
would be closed!!?? What
Another of Maxwell’s equations!
about electric potential
To decide SIGN of flux, use right hand
difference ΔV=∫E•ds?
r r
d!
E
#
d
s
=
"
$
dt
rule: curl fingers around loop C, thumb
indicates direction for dA.
r r
d! B
$C E # ds = " dt
Example
A long solenoid has a circular cross-section of radius R.
The current through the solenoid is increasing at a steady
rate di/dt. Compute the electric field as a function of the
distance r from the axis of the solenoid.
R
The electric current produces a magnetic field B=µ0ni, which changes with time, and
produces an electric field.The magnetic flux through circular disks Φ=∫BdA is
related to the circulation of the electric field on the circumference ∫Eds.
First, let’s look at r < R:
2 dB
E (2! r ) = (! r )
dt
di
2
= (! r ) µ0 n
dt
µ0 n di
E=
r
2 dt
Next, let’s look at r > R:
dB
E (2!r ) = (!R )
dt
2
2
µ0 n di R
E=
2 dt r
electric field lines
magnetic field lines
Example (continued)
µ0 n di R 2
E=
2 dt r
µ0 n di
E=
r
2 dt
E(r)
magnetic field lines
r
r=R
electric field lines
Summary
Two versions of Faradays’ law:
– A varying magnetic flux produces an EMF:
d! B
EMF = "
dt
– A varying magnetic flux produces an electric
field:
r r
d! B
$C E # ds = " dt
Inductors: Solenoids
Inductors are with respect to the magnetic field what
capacitors are with respect to the electric field. They
“pack a lot of field in a small region”. Also, the
higher the current, the higher the magnetic field they
produce.
Capacitance → how much potential for a given charge: Q=CV
Inductance → how much magnetic flux for a given current: Φ=Li
Using Faraday’s law:
di
EMF = ! L
dt
Tesla " m 2
Units : [ L] =
! H (Henry)
Ampere
Joseph Henry
(1799-1878)
“Self”-Inductance of a solenoid
• Solenoid of cross-sectional
area A, length l, total number
of turns N, turns per unit
length n
• Field inside solenoid = µ0 n i
• Field outside ~ 0
i
! B = NAB = NAµ 0 ni = Li
2
N
L = “inductance” = µ 0 NAn = µ 0
A
l
di
EMF = ! L
dt
Example
• The current in a 10 H inductor is
decreasing at a steady rate of 5 A/s.
• If the current is as shown at some
instant in time, what is the
magnitude and direction of the
induced EMF?
(a) 50 V
(b) 50 V
i
• Magnitude = (10 H)(5 A/s) = 50 V
• Current is decreasing
• Induced emf must be in a direction
that OPPOSES this change.
• So, induced emf must be in same
direction as current