DC Circuits
P10-
Examples of Circuits
P10- 2
Current: Flow Of Charge
Average current Iav: Charge Q
flowing across area A in time t
Q
I av 
t
Instantaneous current:
differential limit of Iav
dQ
I
dt
Units of Current: Coulombs/second = Ampere
P10- 3
Direction of The Current
Direction of current is direction of flow of pos. charge
or, opposite direction of flow of negative charge
P10- 4
Clicker question 1
Current Density J
J: current/unit area
 I
J  Î
A
Î points in direction of current

I   J n̂dA   J  dA
S
S

P10- 6
Why Does Current Flow?
If an electric field is set up in a conductor, charge will
move (making a current in direction of E).
Negative charges will move opposite to E’s direction,
The resulted current is still in the same direction as E.
Note that when current is flowing, the conductor is
not an equipotential surface (and Einside ≠ 0)!
P10- 7
Microscopic Picture
Drift speed is velocity forced by applied electric field
in the presence of collisions.
It is typically 4x10-5 m/sec, or 0.04 mm/second!
To go one meter at this speed takes about 10 hours!
How Can This Be?
P10-10
Conductivity and Resistivity
Ability of current to
flow depends on
density of charges &
rate of scattering
Two quantities summarize this:
: conductivity
: resistivity
P10-9
Microscopic Ohm’s Law


J   E


or E   J
or
1

 and  depend only on the microscopic properties
of the material, not on its shape
If current density j is uniform and perpendicular to the
cross-section area, I=jA
P10-10
Temperature Effects on 
For metals, the resistivity is extremely low, because there
are a lot of free charges. The resistivity linearly
increases with increasing temperature over a large
range of T: Because at higher T, the free charges will be
blocked more by the active lattices of atoms.
Semiconductors, such as silicon, have little free charges.
When temperature increases, more electrons can be set
free, hence the conductivity increases and resistivity
decreases.
P10-11
PRS Questions:
Resistance?
P10-12
Why Does Current Flow?
Instead of thinking of Electric Field, think of potential
difference across the conductor
P10-13
Ohm’s Law
V IR
I=V/R; Current is determined by the potential difference V
and the resistance R between any two points.
R= V/I;
The electrical resistance of a circuit
component is the potential difference V between its ends
divided by the current I, that is established in response to the
potential difference.
R
R has units of Ohms () = Volts/Amp
L
P10-15
Electrical Power
Power is change in energy per unit time
So power to move current through circuit elements:
d
d
dq
P  U  qV  
V
dt
dt
dt
P  I V
P10-16
Power - Battery
Moving from the negative to positive terminal of a
battery increases your potential. If current flows
in that direction the battery supplies power
I
Psupplied  I V  I 
P10-17
Power - Resistor
Moving across a resistor in the direction of current
decreases your potential.
Resistors always dissipate power whenever there is
current and convert electrical potential energy into heat.
This is also called Ohmic losses or Joule heating.
Power - Resistor
When the potential drop across the resistor, V, is fixed,
the rate of Joule heating is reversely proportional to R.
When the current through the resistor, I, is fixed, the rate
of Joule heating is proportional to R.
Please notice that regular metal wires has very little
resistance comparing to the actual resistors we use in a
circuit. In general, we consider there is no resistance
and voltage drop in wires, even when a current flows.
PRS Questions:
More Light Bulbs
P10-41
Symbols for Circuit Elements
Battery
Resistor
Capacitor
Switch
P10-21
Sign Conventions - Battery
Moving from the negative to positive terminal of a
battery increases your potential
V  Vb Va
Think:
Ski Lift
P10-22
Sign Conventions - Resistor
Moving across a resistor in the direction of current
decreases your potential
V  Vb Va
Think:
Ski Slope
P10-23
Sign Conventions - Capacitor
Moving across a capacitor from the negatively to
positively charged plate increases your potential
V  Vb Va
Think:
Ski Lodge
P10-24
Series vs. Parallel
Series
Parallel
P10-25
Resistors In Series
The same current I must flow through both resistors
V  I R1  I R2  I (R1  R2 )  I Req
Req  R1  R2
P10-26
Resistors In Parallel
Voltage drop across the resistors must be the same
V  V1  V2  I1 R1  I 2 R2  IReq
V V V
I  I1  I 2 


R1
R2
Req
1
1
1
 
Req R R
1
2
P10-27
PRS Questions:
More Light Bulbs
P10-41