Lecture 26
• Energy and Power
• Resistors in Series
• Real batteries
• Resistors in Parallel
• Example of Resistor Circuits
• Grounding
Energy and Power
•
•
In battery: chemical energy
•
In resistor: work done on charges qEd
kinetic (accelerate) between collisions
thermal energy of lattice after collisions
After many collisions over length L of resistor:
∆Eth = qEL = q∆VR
potential...of charges:∆U
= q∆Vbat = qE
power (rate at which energy supplied to charges): Pbat =
dU
dt
=
dq
dt E
rate at which energy is transferred from current to resistor:
dq
dEth
PR = dt = dt ∆VR = I∆VR
For basic circuit: PR = Pbat (energy conservation)
Using Ohm’s law:
Echem → U → K → Eth → light...
• Common units: PR kW in ∆t hours → PR ∆t kilowatt hours
Example on electric power
•
A standard 100 W (120 V ) lightbulb contains a 9.00!cm-long
tungsten filament. The high-temperature resistivity of tungsten is
-7
9.0 x 10 ohm-meter. What is the diameter of the filament?
Resistors in
Series
•
•
•
current same in each resistor: ∆V1 = IR1 ; ∆V2 = IR2 ⇒ ∆Vab = I (R1 + R2 )
equivalent resistor: Rab =
∆Vab
I
=
I(R1 +R2 )
I
= R1 + R2
Ammeters
measures current in
circuit element
(placed in series):
Req = Rload + Rammeter
ideal, Rammeter = 0 ⇒ current not changed
•
Real Batteries
terminal (user)
voltage
∆ Vba t = E − I r
•
for resistor
connected:
I=
E
Req
=
E
R+r
R
∆VR = IR = R+r
E
∆Vbat = E − Ir;
∆VR = ∆Vbat "= E
•
replace high by
zero resistance
•
maximum
possible current
battery can
produce
Short Circuit
I s h o r t = Er
( ∞ for ideal, r = 0)
•
Parallel Resistors
potential differences same:
∆ V 1 = ∆ V 2 = = V cd
•
•
Kirchhoff’s junction law: I = I 1 + I 2
Ohm’s law:
I=
•
•
•
∆V1
R1
+
∆V2
R2
= ∆Vcd
!
1
R1
+
1
R2
Replace by equivalent resistance:
"−1
!
∆ Vcd
1
1
Rcd = I = R 1 + R 2
"
Identical resistors( R 1 = R 2 ): R se r i es eq = 2R; R p a r a l l e l
In general, R eq < R 1 or R 2 ...in parallel
eq
=
R
2
•
•
•
measure potential difference across circuit
element (place in parallel; don’t need to
break connections, unlike for ammeter)
total resistance: R eq =
1
R
+
1
−1
R volt m eter
in order not to change voltage:
Rvoltmeter ! R (ideally, ∞)
Strategy for Resistor Circuits
•
•
assume ideal wires, batteries
•
•
reduce to equivalent resistor (basic circuit)
use Kirchhoff’s laws and rules for series
and parallel resistors
rebuild using current same for series and
potential difference same for parallel
Voltmeters
Example of
Resistor Circuit
•
Find I and ∆ V for each resistor
•
so far, potential difference, ∆ V (physically
meaningful), enters Ohm’s law (no need to
establish zero point)
•
need common reference point for
connecting 2 circuits
•
grounding: connect 1 point in circuit to
earth ( V e a r t h = 0) by ideal wire: no return
wire (no current); wire is equipotential
point in circuit has earth’s potential
specific value of potential at each point
•
does not change how circuit functions...
unless malfunction/breaking of circuit
(enclose circuit in grounded case insulated
from circuit): case comes in contact with
circuit (
touching would have caused
electrocution) current thru’ wire (fuse
blows...)
Grounding