How much current will flow in a circuit
Analogy:
Want to predict I
Potential difference ∆V
defines amount of the
current that flows in
a circuit
• If water height difference (potential
energy) is maintained, the flow of
water will not change
• The flow of water will depend on the
water height difference
Ohm’s law
Ohm’s law
Current through a conductor is proportional to ∆V:
I∝ ∆V
Notes:
• This is not true for all conductors
• Ohm’s law is not a universal law of physics
(strictly speaking, it is not a law)
Ohm’s law, however, is very useful since it is
simple and works well for many commonly
used materials.
George Ohm
1789-1854
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Resistance: definition
The proportionality coefficient in Ohm’s law is called Resistance
Definition of resistance
∆V
R=
I
Units: Ohm, 1 Ω ≡ 1 V/A
Ohm’s law: I = ∆V
R
Materials that obey Ohm’s law are called ohmic materials.
Ohmic conductor:
tungsten wire
Non-ohmic materials:
fluorescent light tube
semicondutor diode
Resistance: dependence on shape and size
Resistance of a wire depends on its shape, size and properties of
the material it is made of.
Analogy: water pipes
Which one has the highest resistance?
- the one with the smallest area
Which one has the highest resistance?
- the one which is longer
R∝
L
A
2
Resistivity
Resistance of a wire: R = ρ L
A
A
L
L - length
A - cross-sectional area
ρ - resistivity. Units: Ω.m
• The resistivities of good conductors are small.
Resistivities of some materials
Silver
Copper
Aluminum
Nichrome
Carbon
Conductors
1.59×10-8 Ω.m
1.67×10-8 Ω.m
2.65×10-8 Ω.m
108×10-8 Ω.m
3500×10-8 Ω.m
Glass
Rubber
Wood
Insulators
1010-1014 Ω.m
1013-1016 Ω.m
108-1011 Ω.m
Resistivity: dependence on T
Resistivity does not depend on size of shape, it is characteristic of
the material
Resistivity depends on two main factors:
• the number of free charges per unit volume
• the rate of collisions with ions
The rate of collisions depends on the temperature T:
higher T leads to higher ρ:
ρ = ρ0 (1 + α∆T )
ρ0 - resistivity at temperature T0
ρ - resistivity at temperature T=T0+∆T
α - temperature coefficient of resistivity
This dependence
is used in some
thermometers
See page 646 for values of α (p. 635 in old book)
3
“In series” and “in parallel” arrangement
“In series”
When two or more electrical devices
are wired so that the same current
flows through each one
“In parallel”
When two or more electrical devices
are wired so that the potential
difference across them is the same.
Current splits between the devices at
input and then joins back at output.
Resistors in series
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Resistors in series
Loop: DABCD
∆V = 0
∆VDA = VA - VD = ∆Vbattery = E =1.5 V
∆VAB = VB - VA = -IR1
∆VBC = VC - VB = -IR2
∆VCD = VD - VC = 0
1.5 V - IR1 -IR2 = 0
1.5 V - I(R1 + R2) = 0
Requivalent = R1 + R2
1.5 V - IReq = 0
For any number N of resistors
connected in series
Req =
Ri = R1 +R2 + ... + RN
Internal resistance of a battery
Analogy:
People can carry water only so fast, even if we remove the resistance
the water flow will be limited by resistance in the battery
8
Internal resistance of a battery
Real battery can be modeled as an ideal battery in series with a
resistor r.
r - internal resistance of a battery
Part of voltage then drops on r.
The voltage on battery terminals is: ∆V = E − Ir
• ∆V on a load drops as current increases
• Maximum current (Rload=0) is limited by r: Ishort circuit= E/r
• With no load I=0, and ∆V on terminals is equal to emf: ∆V =E
Batteries in series
For any number N of batteries
connected in series
E eq =
req =
E i = E 1 +E 2 + ... + E N
ri = r1 +r2 + ... + rN
• Batteries are connected in series to increase emf.
9
Capacitors in series
Q = C∆V
Symbol:
• Charge Q on all plates is the same
∆VAB = − Q C1
E =
∆VbC = − Q C2
∆VcD = − Q C3
equivalent circuit
1
1
1
1
=
+
+
Ceq C1 C2 C3
Q Q Q
+
+
C1 C2 C3
E =Q
1
1
1
+
+
C1 C2 C3
E =Q
1
Q
=
Ceq
Ceq
For any number N of capacitors
connected in series
1
=
Ceq
1
1
1
1
=
+
+ ... +
Ci C1 C2
CN
Resistors in parallel
Analogy: equivalent resistance should be smaller than any of
the resistances connected in parallel
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