1200 Physics Lecture 2 - F2011
Current and Current density Resistivity and Resistance Power dissipation
Capacitance Calculating capacitance Energy storage
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Current and Current Density Resistivity and Resistance Ohms Law Power dissipation
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Electric Current
Charge flow is produced when an electric field is applied to charges. Positive charges flow in the direction of the electric field, negative charges move opposite to the field. E Current is defined as the rate of flow of charge:
vd
dq i dt
Current flows in the direction that positive charge moves (and opposite to the flow of negative charge.) Current always flows from high potential to low, in the direction of the field.
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Current Density: J
Current density: The rate of current flow per unit area through a cross section to the flow.
conductor +q i
i J or i J dA A area J nqvdrift vdrift drift velocity n carrier density q charge per particle
vA
J
(Current is the flux of current density)
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Kirchhoffs Junction Rule
Because electric charge is conserved, the current through any closed surface must always add up to the change of charge enclosed. At any point in an electrical circuit where charge density is not changing in time, the sum of currents flowing towards that point is equal to the sum of currents flowing away from that point.
I2
dQencl J dA dt J dA 0 in steady state.
P2F09
I1
wires
I3
5
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I1=I2+I3
iClicker Question 3.1
Across which line above is the current the greatest? A=aa B=bb C=cc D=They are the same. E. None of the above. I am a maverick.
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iClicker Question 3.2
Through which line above is the current density the greatest? A=aa B=bb D=They are the same. E. N. o. t. A. (maverick)
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Resistivity
Application of an electric field causes charges to accelerate. Charges scatter off of imperfections in the material through which they move, leading to a drift velocity, which is proportional to the field.
vdrift E
The resistivity is the scaling factor between applied field and current density.
J
E
resistivity
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iClicker Question 3.3
Across which line above is the electric field the greatest? A=aa B=bb D=They are the same E. NotA(Mav)
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Resistance* and resistivity
Resistance, R, is a property of a device, for example, a wire. It depends on shape. Resistivity, , is a property of the material. L R A for a uniform wire with cross section A * is futile and length L
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Resistance and Ohms Law
V=iR
When R is a constant, then this is known as Ohms Law If R varies with voltage, Ohms Law is not obeyed by that material under those conditions. Ohms Law does not hold for light bulb filaments, diodes, transistors
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iClicker Question 3.4
A wire 100 m long and 1 mm in diameter has resistance of 1000 . What is the resistance of a wire of the same material that is 1000 m long and 0.5 mm in diameter? a) 40,000 b) 20,000 c) 1000 d) 250 e.) Mav
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Resistivity of Metals
Resistivity depends on material parameters: Most Important: The density of electrons that can move easily through the material. (~chemical valence and structure) More electrons=lower resistivity. Also important: The ease with which the most mobile electrons move through a material (structure and temperature). For many metals, resistivity increases with temperature due to increased scattering.
0 0 (T T0 )
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Power Dissipation
The rate at which work can be done by charge flowing in a circuit can be deduced from conservation of energy.
dU dqV iVdt dU Power: P iV dt V2 Resistive dissipation: P i 2 R R
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iClicker Question 3.5
A current of 2 A flows through a resistance of 5 ohms. What is the power dissipated in the resistor? A)2 W B)5W C)10W 2 D)20W Pi R E) huh?
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Electric Energy Storage
Capacitors
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What Is a Capacitor?
A device that stores charge when a potential difference is applied The ratio of charge stored to potential difference maintained is the capacitance of object
V+
q(Coulomb) the C(Farad) V (Volt)
-q
q is the magnitude of the charge on one plate
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V
+q
Its worksheet time!
Do section 1 of your worksheet.
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iClicker Question 3.6
S1
A
VB
A capacitor and light bulb are wired in series so that charge flowing into the capacitor must flow through the light bulb. What happens to the brightness of the light bulb after switch S1 is closed powering the circuit? A)It lights briefly. B)It lights and stays on. C)Nothing. It remains off.
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It lights while current flows to charge the capacitor, then stops when the capacitor is charged.
iClicker Question 3.7a
S1
A capacitor and light bulb are wired in series so that charge flowing into the capacitor must flow through the light bulb.
A
VB
v
What happens to the potential across the capacitor when switch 1 is closed? A)It jumps up briefly, then decays. B)It is zero. C)It remains constant. D)It grows to a constant value.
Once the capacitor is charged, the potential across it is is VB.
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iClicker Question 3.7b
If S1 is held in position A for some time, the potential across the capacitor becomes constant. What happens to the potential across the capacitor when S1 is switched to position B? A)It decays to zero slowly. B)It is zero. C)It remains constant. D)It decays to zero immediately.
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A
S1 B
VB
Charge leaves the capacitor by flowing through the bulb.
The RC Time constant
We can write a simple relationship between current and voltage as a capacitor discharges through a resistor.
Combine v iR and q Cv dq dv i C dt dt dv dt dv v RC dt RC v
v(t ) V0e t / RC
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Worksheet Question 3.8a
If S1 is held in position A for some time, the potential across the capacitor becomes constant. What happens to the brightness of the bulb S1 is switched to position B? A)It decays to zero slowly. B)It is zero. C)It remains constant. D)It decays to zero immediately.
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A
S1 B
VB
Charge leaves the capacitor by flowing through the bulb.
Calculating Capacitance: Parallel Plate C
1) Given a charge, find the electric field between the two capacitor plates. 2) Find the potential difference by integrating the field.
Consider a pair of plates with charge density , and area A, separated by distance d .
+ + + + + + + -
C Q /V ;
E from Gauss' law 0
V Ed from V=- E ds V
Q A
A d Qd C 0 0 0 A d
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iClicker Question 3.9
A parallel plate capacitor has two square plates of side length d, separated by distance b. The capacitance is C0. What would the capacitance of a similar capacitor be if the side length were increased to 2d and the distance were increased to 2b? A d2 a) 8C0 C1 d b b) 2C0 4d 2 c) C0/2 C2 2C1 2b d) C0/8 e) There is not enough information to solve for the new capacitance.
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Capacitance: Spherical capacitor
Einside
S
n
q 4 o r
2
dr V Edr cos180 4 o r 2 b q 1 q 1 1 r 4 o b 4 o a b q q ab C 4 o q 1 1 V ba 4 o a b
a
q
a
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Capacitance of our Van de Graaf Sphere
C 4 0 R and for our Van de Graaf R~0.15m, so C 17 pF so for 100000 V, the charge stored is 2 10-6 Coulombs
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Capacitors in Parallel V is the same for each capacitor
q1 = C1V
q2 = C2V
Total charge = q = q1 + q2
q (C1 C2 )V C V V C C1 C2
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Capacitors in Series q is the same
q V1 C1 q V2 C2
q q V V1 V2 C1 C2
Another result: V1 C2 V2 C1
V 1 1 1 q C C1 C2
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Capacitors Store Energy
U qV
1 2
or, since
1 2 2
q CV
2
q U CV 2C
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iClicker Question 3.10
A certain capacitor stores 1J of energy when 100 V is applied across it. How much energy is stored if the voltage is increased to 200 V? A) J D) 2 J B) J E) 4 J C) 1J
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Energy is stored in the Electric Field!
The energy in a capacitor is stored in the electric field between the plates. Ad is the volume between the plates, so 2 1 u 2 0E is the energy per volume in the electric field between the plates. This result is true for all electric fields.
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