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Charging a Capacitor
(instantaneous application of Kirchhoff’s
rules to non-steady-state situation)
Use lower case v, i, q to denote time-varying voltage, current and charge
  q  iR  0
C
t  0: q  0
dq 
q
i
 
dt R RC
Initial current I 0 

R
Final conditions, i=0 Q f  C
dq 
q
 
dt R RC
dq
dt

q  C
RC
i
t
q(t )  C (1  exp(
))
RC
dq 
t
t
i
 exp(
)  I 0 exp(
)
dt R
RC
RC
Time-constant
  RC
When time is small, capacitor charges quickly.
For that either resistance or capacitance must be
small (in either case current flows “easier”)
Discharging a capacitor
q
 IR  0
C
t  0: q  Q
dq
q
I 

dt RC
t
)
RC
Q
t
I (t )  exp(  )
q (t )  Q exp( 


Power distribution systems
Everything is connected in parallel
V=120 V (US and Canada)
V=220-240 V (Europe, Asia)
Circuit Overloads and Short Circuits
Fuse
Circuit breaker
Utility power (kW*h)
1 kW  h  (103W )(3600s)  3.6  106 J
Magnetism
First observation ~2500 years ago
in fragments of magnetized iron ore
Previously, interaction was thought
in terms of magnetic poles
The pole that points North on the magnetic
field of the Earth is called north pole
When points South – south pole
By analogy with electric field bar magnet
sets up a magnetic field in a space around it
Earth itself is a magnet. Compass needle
aligns itself along the earth’s magnetic field
Earth as a magnet
Magnetic Poles vs Electric Charge
The interaction between magnetic poles is similar to the Coulomb
interaction of electric charges BUT magnetic poles always come in
pairs (N and S), nobody has observed yet a single pole (monopole).
Despite numerous searches, no evidence of magnetic charges exist.
In other words, there are no particles which create a radial magnetic
field in the way an electric charge creates a radial field.
Magnetic Field
Electric charges produce electric fields E and, when
move, magnetic fields B
In turn, charged particles experience forces in those
fields:
Lorentz force acting on charge q moving with velocity
v in electric field E and magnetic field B
F  q (E  v  B )
For now we will concentrate on how magnetic force affects
moving charged particles and current-carrying conductors…
Like electric field, magnetic field is a vector field, B
Magnetic Forces on Moving Charges
Force F is perpendicular to the plane of v and B
and numerically equal to
F  q v B  q vB sin 
Direction of F is specified as follows

 
F  q v B
Unit for magnetic field :
N
N

C  m/s A  m
1 T  10 4 Gauss (G)
1 Tesla (T) 
The right hand rule is a useful mnemonic for visualizing the direction of a magnetic force
as given by the Lorentz force law. The diagrams above are two of the forms used to
visualize the force on a moving positive charge. The force is in the opposite direction for
a negative charge moving in the direction shown. One fact to keep in mind is that the
magnetic force is perpendicular to both the magnetic field and the charge velocity, but that
leaves two possibilities. The right hand rule just helps you pin down which of the two
directions applies.
Measuring Magnetic Fields with Test Charges
Total force with both electric and
magnetic fields acting on the charge q


Example: Magnetic force on a proton
Beam of protons moves at v=300000 m/s
through a uniform field B=2.0 T at an angle
30 degrees relative to the field direction
 
F  q( E  v  B )
Alternative rule – direction of right-hand-thread
screw would advance when turned in the same
direction as rotation of vector v toward B for a
positive charge

B
-

v
Which direction does the charge deflect?
a)Up
b)Down
c)It keeps going straight
Magnetic field does NO work; only the
direction of the velocity changes, not its
magnitude!
Application: The Mass Spectrometer
An atom or molecule is ionized by knocking one or
more electrons off to give a positive ion. This is true
even for things which you would normally expect to
form negative ions (chlorine, for example) or never
form ions at all (argon, for example). Mass
spectrometers always work with positive ions.
The ions are accelerated so that they all have the same
kinetic energy.
The ions are then deflected by a magnetic field
according to their masses. The lighter they are, the more
they are deflected.
The amount of deflection also depends on the number
of positive charges on the ion - in other words, on how
many electrons were knocked off in the first stage. The
more the ion is charged, the more it gets deflected.
The beam of ions passing through the machine is
detected electrically.
Magnetic Field Lines
IMPORTANT – Magnetic field lines are NOT
lines of force !!!
The force is always perpendicular
to magnetic field lines.
Sources of Magnetic Fields and Field Lines
If magnetic fields exert forces on moving electric charges, then moving electric charges
create magnetic fields, i.e. currents produce fields.
Follows right-hand rule: point thumb of right
hand in direction of current - magnetic field
curls around wire in direction of curled
fingers
If the current flows in a loop,
the magnetic field produced is
like a bar magnetic - curl
fingers of right hand in
direction of current flow north pole is in direction of
thumb.
Sources of Magnetic Fields and Field Lines
The electron spins on its axis, giving rise to a electron
current in the direction of rotation. Think of the electron
as a ball with charge distributed over its surface. When
the ball spins, that charge is set in motion around the
electron's spin axis, resulting in a magnetic field specific
to the electron.
The electron is like a magnetic dipole, a miniature
magnet, with a north end and a south end.
In most substances, electrons spin in random directions - magnetic fields cancel. For iron
and other magnetic substances, the spin magnetism is not canceled. Can be permanently
magnetized by placing in strong magnetic field and permanently aligning atoms - can be
demagnetized by dropping magnet and jostling atoms out of alignment.
Electromagnetic produced by wrapping coil around
iron bar - magnetic field produced that aligns atoms
in bar - more coils or more current - larger magnetic
field and greater atomic alignment