DC Electrical Circuits

1
DC Electrical Circuits

• Chapter 28
• (Continued)
• Circuits with Capacitors

2
Kirchhoffs Laws
The loop method is based on two laws devised by
Kirchoff

• 1. At any circuit junction,
• currents entering must
• equal currents leaving.

I2
I1
I3 I1 I2
I
2. Sum of all DVs across all circuit
elements in a loop must be zero.
r
R

E
-
E - Ir - IR 0
3
RC Circuits
So far we have considered simple circuits with
either capacitors or resistors. Now we will
consider more complicated circuits with both
resistors and capacitors RC Circuits.
The battery pushes current until the capacitor is
fully charged. After this no current flows. (A
small lie.) This problem is time dependent.
R

E
C
-
4
RC Circuits Charging
open
closed
R
I
VRIR


E
- - -
-
VCq/C
C
When the switch closes, at first a high current
flows VR is big and VC is small.
5
RC Circuits Charging
open
closed
R
I
VRIR


E
- - -
-
VCq/C
C
When the switch closes, at first a high current
flows VR is big and VC is small. As q is
stored in C, VC increases. This fights against
the battery so I decreases.
6
RC Circuits Charging
Apply the loop law E IR - q/C 0
Take the derivative of this with respect to time
Now use dq/dt I and rearrange
This is a differential equation for an unknown
function I(t). It is solved subject to the
initial condition I(0) E / R.
7
RC Circuits Charging
?
E
And I(0) I0 E / R ?
8
RC Circuits Charging
E
E
E
E
From this we get
q VC C E C (1 et/RC)
9
Charging
E/R
Current
I
t/RC
E
VC
Potential Drop
VR
t/RC
10
Discharging an RC Circuit
R
q
C
VCV0
-q
Open circuit
Current will flow through the resistor for a
while. Eventually, the capacitor will lose all
its charge, and the current will go to
zero. Power P IV I2R will be dissipated
in the resistor (as heat) while the current
flows.
11
Discharging an RC Circuit
Loop equation q/C - IR 0 ? I q /
(RC)
R
VRIR
q
I
C
VCq/C
Take d/dt ?
-q
Note that I - dq/dt
Here the current at t0 is given by the initial
voltage on the capacitor
I(0) V0/R q0
/RC
This equation is solved very much like the other
(charging case)
12
Discharging an RC Circuit
R
VRIR
q
I
C
VCq/C
-q
The charge on the capacitor is given by q/C
- IR 0 so q C IR q C V
13
Discharging
E/R
Current
t/RC
VR
0
Potential Drop
VC
t/RC
14
Example A capacitor C discharges through a
resistor R. (a) When does its charge fall to
half its initial value ?
Charge on a capacitor varies as
R
Q
C
I
15
Example A capacitor C discharges through a
resistor R. (a) When does its charge fall to
half its initial value ?
Charge on a capacitor varies as
R
Q
C
Find the time for which QQ0/2
I
16
Example A capacitor C discharges through a
resistor R. (a) When does its charge fall to
half its initial value ?
Charge on a capacitor varies as
R
Q
C
Find the time for which QQ0/2
I
17
Example A capacitor C discharges through a
resistor R. (a) When does its charge fall to
half its initial value ?
Charge on a capacitor varies as
R
Q
C
Find the time for which QQ0/2
I
RC is the time constant
18
Example A capacitor C discharges through a
resistor R. (b) When does the energy drop to
half its initial value?
The energy stored in a capacitor is
We seek the time for U to drop to U0/2
19
Example A capacitor C discharges through a
resistor R. (b) When does the energy drop to
half its initial value?
The energy stored in a capacitor is
We seek the time for U to drop to U0/2
20
Magnetic FieldsChapter 29

• Permanent Magnets Magnetic Field Lines
• The Magnetic Force on Charges

21
Magnetism

• Our most familiar experience of magnetism is
  through permanent magnets.
• These are made of materials which exhibit a
  property called ferromagnetism - i.e., they can
  be magnetized.
• Depending on how we position two magnets, they
  will attract or repel, i.e. they exert forces on
  each other.
• Just as it was convenient to use electric fields
  instead of electric forces, here too it is
  useful to introduce the concept of the magnetic
  field B.
• There are useful analogies between electric and
  magnetic fields, but the analogy is not perfect
  while there are magnetic dipoles in nature, there
  seem to be no isolated magnetic charges (called
  magnetic monopoles). And the force laws are
  different.
• We describe magnets as having two magnetic poles
• North (N) and South (S).
• Like poles repel, opposite poles attract.

22
Field of a Permanent Magnet
Shown here are field lines. The magnetic field B
at any point is tangential to the field line
there.
23
Field of a Permanent Magnet
The south pole of the small bar magnet is
attracted towards the north pole of the big
magnet. Also, the small bar magnet (a magnetic
dipole) wants to align with the B-field. The
field attracts and exerts a torque on the small
magnet.
24
Magnetism

• The origin of magnetism lies in moving electric
  charges.
• Moving (or rotating) charges generate magnetic
  fields.
• An electric current generates a magnetic field.
• A magnetic field will exert a force on a moving
  charge.
• A magnetic field will exert a force on a
  conductor that carries an electric current.

25
What Force Does a Magnetic Field Exert on Charges?

• If the charge is not moving with respect to the
  field (or if the charge moves parallel to the
  field), there is NO FORCE.

q
26
What Force Does a Magnetic Field Exert on Charges?

• If the charge is not moving with respect to the
  field (or if the charge moves parallel to the
  field), there is NO FORCE.

q

• If the charge is moving, there
• is a force on the charge,
• perpendicular to both v and B.
• F q v x B

q
27
Force on a Charge in aMagnetic Field
F
v
q
m
B
(Use Right-Hand Rule to determine direction of
F)
28
Units of Magnetic Field
Since
Therefore the units of magnetic field are
(Note 1 Tesla 10,000 Gauss)
29
The Electric and Magnetic Forces are Different
Whereas the electric force acts in the same
direction as the field
The magnetic force acts in a direction orthogonal
to the field
30
The Electric and Magnetic Forces are Different
Whereas the electric force acts in the same
direction as the field
The magnetic force acts in a direction orthogonal
to the field
(Use Right-Hand Rule to determine direction of
F)
31
The Electric and Magnetic Forces are Different
Whereas the electric force acts in the same
direction as the field
The magnetic force acts in a direction orthogonal
to the field
(Use Right-Hand Rule to determine direction of
F)
And the charge must be moving.
32
Trajectory of Charged Particlesin a Magnetic
Field
(B field points into plane of paper.)
B

v
F
33
Trajectory of Charged Particlesin a Magnetic
Field
(B field points into plane of paper.)
v
B
B


v
F
F
34
Trajectory of Charged Particlesin a Magnetic
Field
(B field points into plane of paper.)
v
B
B


v
F
F
Magnetic Force is a centripetal force
35
Rotational Motion
? s / r ? s ? r ? ds/dt d?/dt r ? v
? r
s
?
r
? angle, ? angular speed, ? angular
acceleration
at r ? tangential acceleration ar v2 /
r radial acceleration
?
at
ar
The radial acceleration changes the direction of
motion, while the tangential acceleration changes
the speed.
Uniform Circular Motion
? constant ? v and ar constant but direction
changes
ar
?
KE (1/2) mv2 (1/2) mw2r2
ar v2/r ?2 r
v
F mar mv2/r m?2r
36
Radius of a Charged ParticleOrbit in a Magnetic
Field
Centripetal Magnetic Force
Force

37
Radius of a Charged ParticleOrbit in a Magnetic
Field
Centripetal Magnetic Force
Force

v
B

F
r
38
Radius of a Charged ParticleOrbit in a Magnetic
Field
Centripetal Magnetic Force
Force

39
Radius of a Charged ParticleOrbit in a Magnetic
Field
Centripetal Magnetic Force
Force

v
B

F
r
Note as , the magnetic force does
no work.
40
Cyclotron Frequency
The time taken to complete one orbit is
41
Cyclotron Frequency
The time taken to complete one orbit is
Hence the orbit frequency, f
42
Cyclotron Frequency
The time taken to complete one orbit is
Hence the orbit frequency, f
- known as the cyclotron frequency
T 2?/? 1/ƒ ? ƒ ?/2?
43
The Electromagnetic Force
If a magnetic field and an electric field are
simultaneously present, their forces obey the
superposition principle and must be added
vectorially
The Lorentz force


q
44
Exercise
electron
B
v
v

• In what direction does the magnetic field
  point?
• Which is bigger, v or v ?

45
Exercise answer
electron
B
v
v
F

• In what direction does the magnetic field point
  ?
• Into the page F -e v x B
• Which is bigger, v or v ?
• v v B does no work on the electron, F?v

46
What is the orbital radius of a charged particle
(charge q, mass m) having kinetic energy K, and
moving at right angles to a magnetic field B, as
shown below?.
x
x
x
B
x
x
x
K

q m
47
What is the orbital radius of a charged particle
(charge q, mass m) having kinetic energy K, and
moving at right angles to a magnetic field B, as
shown below?.
F q v x B m a and a v2 / r
q v B m v2 / r
x
x
x
B
x
x
x
q B m v / r ? r q B m v
r
r m v / (q B)
K (1/2) mv2

q m
r2 m2 v2 / (q B)2
(1/2m) r2 K / (q B)2 ? r 2mK1/2 / (q B)
48
What is the relation between the intensities of
the electric and magnetic fields for the
particle to move in a straight line ?
The magnetic field points into the picture. The
direction of the electric field is not yet
specified.
49
What is the relation between the intensities of
the electric and magnetic fields for the
particle to move in a straight line ?
FE q E and FB q v B
If FE FB the particle will move following a
straight line trajectory
q E q v B
FE
FB

50
What is the relation between the intensities of
the electric and magnetic fields for the
particle to move in a straight line ?.
x
x
x
B
E
FE q E and FB q v B
x
x
x
If FE FB the particle will move following a
straight line trajectory
v

q m
q E q v B
FE
FB

So need E pointing to the right.
51
Trajectory of Charged Particlesin a Magnetic
Field
What if the charged particle has a velocity
component along B?
unchanged
Circular motion in xy plane.
x
z
y