Today in Physics 122: review of DC circuits,
magnetostatics,
i and
d induction
i d i
Shanghai’s highspeed maglev
train, leaving the
airport (Shanghai
Metro).
8 November 2012
Physics 122, Fall 2012
1
DC circuits: combinations of Rs, Cs
Parallel and series pairs of resistors and capacitors:
C1
=
C2
C1
C2 =
8 November 2012
1
1
1


C eq C1 C 2
C1C 2
C eqq 
C1  C 2
 C1 , C 2
C eq  C1  C 2
 C1 , C 2
R1
=
R2
Req  R1  R2
 R1 , R2
1
1
1


Req R1 R2
R1
Physics 122, Fall 2012
R2 =
R1 R2
Req 
R1  R2
 R1 , R2
2
DC circuits: Kirchhoff
Kirchhoff’ss Rules
1. At each node – the
conductor of
whatever shape that
j i two
joins
t
or more
components – the
current flowing in is
equal to the current
flowing out.
2. Voltages along any
complete loop in a
circuit must add
dd up
to zero.
8 November 2012
I1
-
+
R1
+
- V0
dQ dt
+
-
I2
R2
+
C
-
t0
I 1  I 2  dQ dt  0
V0  I 1 R1  I 2 R2  0
I 2 R2  Q C  0
Physics 122, Fall 2012
3
Kirchhoff’ss Rule recipe
Kirchhoff
 Define a current – value and
direction – through each
component in the circuit.
• Current (dQ/dt) flows into
+ end of C, from + end of
V and +V,
+ in R.
R
 In the node rule (KR#1),
bookkeep all currents flowing
into the node as positive, and
g out as
all those flowing
negative; the sum of all the
positives and negatives comes
tto zero.
8 November 2012
Physics 122, Fall 2012
I1
-
+
R1
+
- V0
dQ dt
+
-
I2
R2
+
C
-
t0
I 1  I 2  dQ dt  0
4
Kirchhoff’ss Rules recipe (continued)
Kirchhoff
 In the loop rule (KR#2), count
voltages traversed from - to +
as positive, and those
t
traversed
d from
f
+ tto – as
negative.
• That is,
is voltage drops
across a resistor if you’re
following
g voltage
g in the
direction the current flows
through it.
I1
-
+
R1
+
- V0
dQ dt
+
-
I2
R2
+
C
-
t0
I 1  I 2  dQ dt  0
V0  I 1 R1  I 2 R2  0
I 2 R2  Q C  0
8 November 2012
Physics 122, Fall 2012
5
Kirchhoff’ss Rules recipe (continued)
Kirchhoff
 Identify the unknown quantities – N, say – in the circuit,
and count them.
 Then write the node rule and/or the loop rule to generate
as many relations between the voltages and currents as
there are unknowns (N).
• Use
U both
b th th
the node
d lloop rules,
l att least
l t once each.
h
 This gives a system of N equations in N unknowns, either
purely linear equations (if only R and V),
V) or first
first-order
order
differential equations (if there are Cs). Solve them.
 Important point: it doesn
doesn’tt matter whether you correctly
guess the direction of each current. If you guess wrong,
your answer will jjust be a negative
y
g
number.
8 November 2012
Physics 122, Fall 2012
6
Magnetostatics
Force laws:
dF  Id   B  IdB sin  nˆ

currents

F  Id   B  IB sin  nˆ d 

F  Qv  B
point charges


Cross products:
a  b  b  a  ab sin  nˆ
 a  b  0 if a  b
 a  b  abnˆ if a  b
Cross products of Cartesian
unit vectors:
xˆ  yˆ  zˆ yˆ  zˆ  xˆ zˆ  xˆ  yˆ
8 November 2012
Physics 122, Fall 2012
Id  or
Qv
Qv
B
dF or
F

Lay
y fingers
g
of right
g hand along
g
arc from d to B, pointing to B.
Thumb points along dF.
7
Magnetostatics (continued)
r
 r
Field laws:
0
B
4


Id   r
 r
r  r
2

r
BiotSavart
field law
r
 Bd  0 Iencll
r  r
Id
dB

 0

I
J dA
Id

where 0  4  10 7 T m A 1
8 November 2012
Physics 122, Fall 2012
Or: th
O
thumb
b along
l
Id ; fingers curl
in direction of dB.
8
Magnetostatics (continued)
Field laws:
0
B
4



Id   r
 r
r  r


 J dA

d
2
Bd   0 I encll
 0
B

Ampère’s
law
J

where 0  4  10 7 T m A 1
8 November 2012
Physics 122, Fall 2012
9
Biot-Savart
Biot
Savart field law recipe
The setup, and most of the execution, of B calculations from
the
h Biot-Savart
Bi S
fi ld law
field
l
are the
h same as for
f E calculations
l l i
using Coulomb’s law.
That is,
is
 choose an appropriate coordinate system,
 dissect the source distribution into infinitesimal elements,
 use the symmetry of the source distribution to simplify
the vector addition as much as possible, and then
 integrate the resulting expression.
No new tricks are involved, and no new complications
b id th
besides
the iintrusion
t i off cross products,
d t and
d fi
fields
ld that
th t li
lie
sideways with respect to the distance from the infinitesimal
element.
8 November 2012
Physics 122, Fall 2012
10
Ampère’ss Law recipe
Ampère
Though the dimension of its integration is reduced by one on
both sides of the equation compared to Gauss’s Law,
Ampère’s Law is applied to find B in much the same way that
G
Gauss’s
’ Law
L
is
i applied
li d to
t find
fi d E.
E
Ampère’s law
That is, use the symmetry of the
geometries
current distribution to
Infinite linear current
 determine the direction of B, and
Infinite planar current
 identify closed curves along
Infinite cylindrical
which the magnitude of B is
current, any radial
g which
uniform,, or along
dependence
p
Infinite solenoid
 Then B comes out of the integral;
Toroid
g integrals
g
are
the two remaining
simple, enabling solution for B.
8 November 2012
Physics 122, Fall 2012
11
Magnetic materials
Permeability:   Km 0 .
Diamagnetic (Km < 1; e.g. Cu) and paramagnetic (Km > 1; e.g
Al) materials have small permeability ( Km  1  10 5  10 4 ).
Ferromagnetic materials (e.g. Fe, Ni, Co, Sm) can have very
large permeability: Km ~ 1000.
B   nI  Km 0 nI
Km  1 as
drawn.
I
8 November 2012
Physics 122, Fall 2012
12
Induction and Faraday
Faraday’ss Law
 If the flux of B enclosed by a circuit changes with time, an
electromotive force is induced, that drives a current and
associated additional B, which opposes the change.
 Equivalently:
l l iff the
h fl
flux off B through
h
h a surface
f
changes
h
with time, so does the integral of E around the boundary
of the surface.
surface
Stated instead as equations,

d
d
B
 
dA
dt
Faraday’s
d
L
Law

 Ed   dt  BdA



d
8 November 2012
Physics 122, Fall 2012
13
Faraday’ss Law recipe
Faraday
 Calculate the magnetic flux, and differentiate to get its
rate
t off change.
h
Thi
This ttells
ll you th
the magnitude
it d off th
the
induced EMF right away.
 To get the polarity
polarity, consider how the EMF can produce a
current, and thus its own B, to oppose the change in flux.
• If  BdA is decreasing
decreasing, make I
produce more field in the same
B
direction as the external field.
• If BdA is increasing, make I
produce more field in the
opposite direction as the
external field.
I
• Remember
b which
h h way I produces
d
B. (See
(
ffigure.))

8 November 2012
Physics 122, Fall 2012
14
General advice
 Don’t forget the chain rule of differentiation or the
fundamental theorem of calculus.
• For example, if you know  B  x  and want to know
h
d B dt , note that
d B d B dx d B
vx .


dt
d dt
dx
ddx
• Or: if you know you’re going to have to take the
d i i off a function
derivative
f
i you just
j
produced
d
d by
b
integration, save a step with
x
d
f  x   dx   f  x 

dx
a
8 November 2012
Physics 122, Fall 2012
15
General advice (continued)
 Don’t forget the principle of superposition: a complicated
current, for example, can often be broken down into two
simple ones for which the fields are calculated simply and
th superposed.
then
d Lik
Like uniform
if
currentt iin a cylinder
li d with
ith
an off-center hole:
JI A
=
J
+
J
 Don’t forget the cross products of Cartesian unit vectors
(page 7).
7)
8 November 2012
Physics 122, Fall 2012
16
The upcoming midterm exam
Midterm #2: 8-9:15 AM,, Tuesday
y, 13 November 2012,,
Hubbell Auditorium (141 Hutchison Hall)
 Covers all topics
p reviewed here: those introduced since
the last exam (chapters 26-29 in the book)
 No class on Tuesday after the exam; lectures will resume
on Thursday.
8 November 2012
Physics 122, Fall 2012
17