Lecture 25
• Resistance and Ohm’s law
• chapter 32
Resistance and Ohm’s law
Current created by E (requires ∆V )
•
∆V related to I
• E constant by current conservation
∆V
∆V
E = ∆s = L ; I = JA; J = σE
ρL
A
I = ρL ∆V ; R (resistance) = A
•
⇒
R property of specific wire (depends on material and L, A):
Unit of R: 1 ohm = 1 Ω ≡ 1 V/A
(Resistivity: property of material only)
•
resistors: circuit elements with resistance larger than wires used
to limit current
Batteries and current
•
so far, current from discharge of
capacitor: transient (stops when
excess charge removed)
•
battery: sustained current due
energy from chemical reactions
used to “lift” charge...falls
“downhill” in wire (warms it)
Cause and Effect
• Battery source of ∆V (= E for ideal)
bat
∆Vwire = ∆Vbat (independent of path)
• ∆Vwire causes
charges)
s
(surface
•
E results in current: I = JA = σAE
•
Current determined by both battery
and wire’s R
Ohmic and Nonohmic
materials; Ideal Wire
Model
•
•
•
ideal wires: R = 0 ⇒
∆V = 0 even I != 0
resistors:
10 to 106 Ω
ideal insulators:
R=∞⇒
I = 0 even if ∆V != 0
Chapter 31 (Fundamentals of Circuits)
•
understand fundamental principles of electric circuits; direct
current (DC): battery’s potential difference, currents constant
• Kirchhoff’s Laws and Basic Circuit
• Energy and Power
• Resistors in Series
• Real batteries
• Resistors in Parallel
Circuit Elements and Diagrams
•
circuit diagram: logical picture of connections (replace
pictures of circuit elements by symbols)
Kirchhoff’s Laws
•
circuit analysis: finding potential
difference across and current in
each component
•
junction law (charge conservation)
!
!
Iin = Iout
•
•
loop law (energy conservation)
!
∆Vloop = i (∆V )i = 0
strategy:
assign current direction
travel around loop in direction of current
Vbat = ±E; VR = −IR
apply loop law
Basic Circuit
• junction law not needed
• ideal wires: no potential difference
law: ∆V
= ∆V + ∆V = 0
• loop
∆V = +E;
loop
bat
bat
R
∆VR = Vdownstream − Vupstream = −IR ⇒
E
E − IR = 0; I = R
;∆VR = −IR = −E
A more complex circuit
•
charge can flow “backwards”
thru’ battery: choose cw
direction for current (if solution
negative, current is really ccw)
•
loop law: ∆Vbat 1 + ∆VR1 + ∆Vbat 2 + ∆VR2 = 0
∆Vbat 1 = +E1 ; ∆Vbat 2 = −E2 ;
∆VR1 = −IR1 ; ∆VR1 = −IR2 ⇒
E1 − IR1 − E2 − IR2 = 0 ⇒
6 V - 9 V = = −0.5 A...
2
=
I = RE11 −E
+R2
4Ω+2Ω
(expected: 9 V battery “dictates” direction...)
∆VR1 = −IR1 = +2.0 V...
Energy and Power
•
•
In battery: chemical energy potential...of charges: ∆U
•
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
= 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
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:
E
R eq
E
R+r
I =
=
∆ V R = I R = R R+ r E
∆ Vba t = E − I r ;
∆ V R = ∆ Vba t = = 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:
∆ V2
∆ V1
I = R 1 + R 2 = ∆ V cd
1
R1
+
1
R2
−1
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