Transcript PHYS_3342_101811

Calculating resistance
A variable cross-section resistor treated as a serial
combination of small straight-wire resistors:
a b
r ( x)  b 
x;
h
h
dx 
dr
a b
h
dx
R   dR    2
r ( x)
0
 h
h
 2
dr 
r a  b
ab
b
a
Example: Equivalent resistances
Series versus parallel connection
What about power delivered to each bulb?
P  I 2 R or
P  I 2 R or
Vab2 Vbc2
P

R
R
Vde2
P
R
What if one bulb burns out?
Symmetry considerations to calculate equivalent
resistances
No current through the resistor
All resistors r
Currents : I1  I / 3; I 2  I1 / 2
I2
I1
I1
I1
I2
I2
I2
I2
I1
I1
I1
I2
Total voltage drop between a and b :
1 1 1
5
V  I (   )r  I r
3 6 3
6
5
R r
6
Kirchhoff’s rules
To analyze more complex (steady-state) circuits:
1. For any junction: Sum of incoming currents
equals to sum of outgoing currents
(conservation of charge)
I  0
Valid for any junction
2. For any closed circuit loop: Sum of the voltages
across all elements of the loop is zero
(conservation of energy)
V  0
-
Valid for any close loop
The number of independent equations will be
equal to the number of unknown currents
Loop rule – statement that the electrostatic force is conservative.
Sign conventions for the loop rule
A single-loop circuit
 IR1   2  IR2   1  0


I
1
2
R1  R2
With the numbers given,
I is negative
(Important is only consistenc y)
Charging of a car battery
Complex networks
Find currents, potential differences and equivalent resistance
  I1 (1)  ( I1  I 3 )(1)  0 (1)
  I 2 (1)  ( I 2  I 3 )(2)  0 (2)
I 3  1 A
I1  6
A
 I1 (1)  I 3 (1)  I 2 (1)  0 (3)
I2  5
A
Electrical Measuring Instruments
Galvanometer
Can be calibrated to measure
current (or voltage)
Example: Full-scale deflection
Ifs =1 mA, internal coil resistance
Rc =20 
V  I fs Rc  0.020V
I fs Rc  ( I a  I fs ) Rsh
For max current reading Ia of 50mA
Rsh  0.408 
Req  0.4 
Vv  I fs ( Rc  Rsh )
For max voltage reading Vv =10V
Rsh  9980 
Req  10,000 
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
q (t )  Q exp( 
)
RC
Q
t
I (t )  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