Download Document

Electric Current
and
Direct Current Circuits
Why bother with Electric Potential V,
Capacitance, and Electric Fields?
• The electric potential difference between point a) and point
b) in a circuit is the force driving current from a) to b)
• Capacitance is universal,
 Capacitance limits the speed of switching circuits
 Capacitance stores energy
• Electric Field energy is the energy of waves, light.
2
Electric Current
Whenever there is a net movement of charge, there exists
an electrical current.
A current can flow in a wire: usually electrons.
A current can flow in a liquid solution:
For example Na+, and K+ ions across a nerve cell
membrane.
A current can flow in air or free space:
electron or ion beam, lightning.
3
Unit of measure
of Electric Current
If a charge Q moves through a “surface” A in a time 
t, then there is a current I:
Q
I 
t
The unit of current is the Ampere (A): 1 A= 1
Coulomb/sec.
By convention, the direction of the current is the
direction of flow of the positive charges.
If electrons flow to the left, that is a positive current to
the right.
eI
4
Ohm’s Law
For many materials, the current I is directly proportional to the
voltage difference V.
We define the resistance, R, of such a material to be:
The unit of resistance is Ohms (W): 1 W  1 Volt/Amp
V
R
I
Common resistors used in electrical circuits range from a few W to
MW (106W).
If R is constant: doesn’t depend on current, or history of current
flow, and only small variation with temperature, atmospheric
pressure, etc,
the material is said to be ohmic, and we write Ohm’s Law:
V  IR
5
Fluid Analogy of Resistance
• A fluid (liquid or gas) will not flow through a narrow tube unless there is a
pressure difference between the input and output ends.
Fluid flow
Pin
v
Pout
• The pressure difference can be provided by external pressure, or by gravity.
• The longer the tube, or the narrower the tube, the larger a pressure difference
(or gravity gradient) is required to maintain the same flow.
Liquid flow
6
Resistivity
An object which provides resistance to current flow is called a
resistor. The actual resistance depends on:
• properties of the material (resistivity)
• the geometry (length and cross sectional area)
For a conductor of length L and cross-sectional area A,
the resistance is R= L/A, where  is called the resistivity.
L

Area A
7
Wires & Resistors in Circuits
• A piece of wire is a resistor.
• However, for good conductors like Cu, Al, Au, Ag,
the resistivity is extremely low.
• When we analyze a circuit containing wires and
other elements (such as light bulbs), the resistance
of the wires is so low that we can [usually] pretend
the wires are perfect conductors.
• Current can flow in the wire even though the
potential is everywhere the same inside each
separate piece of wire.
8
Sample Resistivity values
Material
Resistivity 
(Wm)
Silver (Ag)
1.59 10-8
Tungsten (W)
5.6 10-8
Nichrome
100 10-8
Graphite ( C )
350010-8
Si
2.5 10-3
Glass
> 1010
9
Walker problem #8
A silver wire is 4.5 m long and 0.40 mm in diameter. What is its
resistance? 0.569 W
10
Walker problem #14
A typical cell membrane is 8.0 nm thick and has an electrical
resistivity of 1.3x107 W·m.
If the potential difference between the inner and outer surfaces of a
cell membrane is 80 mV, how much current flows through a square
area of membrane 1.3 µm on a side?
11
Resistors in Circuits
• In drawing a circuit, the symbol for a resistor is
• This zigzag pattern is a visual reminder that the
material of the resistor impedes the flow of charge,
and it requires a potential difference V between the
two ends to drive current through the resistor.
• Current flows from higher value of potential to lower
value of potential
12
Simple Battery Circuit
• A battery is like a pump
 A pump raises fluid by a height h.
 A battery pumps charge up to a higher potential.
I
I = V/R
R
V +
0
I
Current is the same everywhere.
Voltage varies from point to
point around loop.
13
An Incandescent Light Bulb is a Resistor
(but R depends on Temperature T of filament, and
T depends on current I).
14
Question: Which Circuit will light the bulb?
A)
B)
C)
15
Power in Electric Circuits
Recall that resistance is like an internal friction - energy is
dissipated. The amount of energy dissipated when a charge
Q flows down a voltage drop V in a time t is the power P:
P =U/ t =(Q·V/t) = IV
SI unit: watt, W= Amp·Volt = C V/s = J/s
For a resistor, P=IV can be rewritten with Ohm’s Law V=IR,
P = I2R = V2/R
Power is not Energy, Power is rate of consumption (or production) of energy
Large power plants produce between 100 MW and 1GW of power. This
power is then dissipated in the resistors and other dissipative circuits in our
electronic appliances, in the resistance of the windings of electric motors, or is
used to charge batteries for later use.
16
Energy and Power
Energy Usage: Power times time = Energy
consumed
1 kilowatt-hour = (1000 W)(3600 s) = (1000 J/s)(3600 s) =
3.6106 J
Electricity in VA costs about $0.15 per KWhr
A typical household uses 1KW of power, on average.
There are ~8800 hours in a year
In one year, each household consumes 8800 KWhr, (8800
KW·hr)($0.15) = $1320.
17
Direct Current (DC) Circuits
A circuit is a loop comprised of elements such as batteries,
wires, resistors, and capacitors through which current flows.
Current can only flow around a loop if the loop is continuous.
Any break in the loop must be described by the capacitance of
the gap, which allows charge to build up as current flows onto
the capacitors.
For current to continue flowing in a circuit with non-zero
resistance, there must be an energy source. This source is
often a battery. A battery provides a voltage difference across its
terminals.
18
Simple Battery Circuit
• An incandescent light bulb can be
approximated as an ideal resistor
(this is a bad approximation,
because most light bulbs have a
very strong temperature
dependence to the resistance).
• V=IR
• 5 Watt bulb with 3 V battery:
• P= V2 / R
• R = V2/P = (3V) 2/(5 A·V)
• R = 1.8 V/A = 1.8 W.
19
Direct Current (DC) Circuits - MORE
• Includes:
batteries, resistors, capacitors
• Kirchoff’s Rules
- conservation of charge
I  0
(junction rule, valid at any junction)
- conservation of energy
(Laws) follow from:
(loop rule, valid for any loop)
V  0
• With emf ():
• “charge pump”
constant current can be maintained
• SI unit for emf
Volt (V)
• No resistance
connecting wires of the loop
forces electrons to move
in a direction opposite to the electric field
20
Kirchhoff’s Rules
• Any charge must move around any closed loop with emf
• Any charge must gain as much energy as it loses
• Loss:
IR – potential drop across resistor
• Gain:
chemical energy from the battery
(charge go reverse direction from )
Often what seems to be a complicated circuit can be reduced to a
simple one, but not always. For more complicated circuits we
must apply Kirchhoff’s Rules:
• Junction Rule: The sum of currents entering a junction equals
the sum of currents leaving a junction.
• Loop Rule: The sum of the potential difference across all the
elements around any closed circuit loop must be zero.
21
Combining Circuit Elements
Any two circuit elements can be combined in two
different ways:
• in series - with one right after the other, or
• in parallel - with one right next to the other.
Series
Combination
Parallel
Combination
22
Resistors in Series
• By the conservation of charge,
 The same current I flows through
all three resistors, and through the
battery.
 Junction rule at a, b, c, d,
separately
b
a
c
d
•Ohm’s law:
+ E – I•R1 – I•R2 – I•R3 = 0
E = I•R1 + I•R2 + I•R3
I•Rtotal = I•(R1+R2+R3)
Rtotal = (R1+R2+R3)
23
Resistors in Parallel
• The current I splits into three (non-equal)
branches such that I=I1+I2+I3.
•Ohm’s law:
•  = (/R1 + /R2 + /R3) Req =  Req (1/R1 + 1/R2 + 1/R3)
1/ Req = (1/R1 + 1/R2 + 1/R3)
24
Resistors in both Series and Parallel
• Combine the first two in parallel to obtain
equivalent resistance R/2.
• Combine three in series to obtain equivalent
resistance R + (R/2) + R = 2.5 R.
25
What is the equivalent resistance of circuit?
1.0 W
1.0 W
3V
2.0 W
1.0 W
26
What is the current in each resistor?
1.0 W
1.0 W
3V
2.0 W
1.0 W
27
What is the power dissipated in each resistor?
1.0 W
1.0 W
3V
2.0 W
1.0 W
28
Circuits containing Capacitors
Capacitors are used in electronic circuits. The symbol
for a capacitor is
We can also combine separate capacitors into one
effective or equivalent capacitor. 2 capacitors can be
combined either in parallel or in series.
Series
Combination
C1
C2
Parallel
Combination
C2
C2
29
Parallel vs. Series Combination
C1
Q
C2
-Q
Q
-Q
Parallel
Series
• charge Q1 , Q2
• charge on each is Q
• total Q=Q1 + Q2
• total charge is Q
• voltage on each is V
• voltage V1 + V2 = V
• Q1=C1V
• Q=C1V1
• Q2=C2V
• Q=C2V2
• Q=CeffV
• Q= Ceff V = Ceff(V1+V2)
• Ceff=C1+C2
• 1/Ceff=1/C1+1/C2
30
RC Circuits
We can construct circuits with more than just resistor, for
example, a resistor, a capacitor, and a switch:
When the switch is closed the current will not remain
constant.
• Capacitor acts as an open circuit: I=0 in branch with capacitor
under study state condition.
31
Capacitor Charging
Lets assume that at time t=0, the capacitor is uncharged, and
we close the switch. We can show that the charge on the
capacitor at some later time t is:
q=qmax(1-e-t/RC)
RC is known as the time constant , and qmax is the maximum
amount of charge that the capacitor will acquire:
qmax=C
32
Capacitor Discharging
Consider this circuit with the capacitor fully charged at time
t=0:
It can be shown that the charge
on the capacitor is given by:
q=qmaxe-t/RC
33