Electric Current and Direct Current Circuits

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Electric Current andDirect Current Circuits

• Physics 112, Prof. C.E. Hyde-Wright
• Spring 2005
• Walker, Chapter 21

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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.

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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.
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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
• 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.

e-
I
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Resistivity
V1
V2
e-
E
V1 gt V2

• In most materials, in order for a steady current
  I to flow there must be an electric field E
  inside the material. For each charge q, the
  electric field E produces a force F qE, this
  causes the charge to accelerate, however, due to
  collisions with the surrounding medium, there is
  a viscous force roughly proportional to the mean
  velocity of the charges F q E-v g. On
  average, the free charges travel at a constant
  velocity proportional to the electric field
  F0, v q E/ g.
• If n is the density of free-charges in the
  material, and A is the cross sectional area of
  the material, then the current flowing is
• I n A q v A(nq2/g)E A E / r r g/(nq2)
• r Resistivity instrinsic property of the
  medium.

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Voltage, Current, and Resistance

• A block of material has resistivity r, cross
  sectional area A, and length L.
• If there is an electric field E in the material,
• current I A E/r will flow
• from higher potential V1 to lower potential V2.
• E (V1-V2)/L
• Define Resistance R r L / A
• V1 - V2 I R Ohms Law.

V1
V2
e-
E
Resistivity ?0
Resistivity r
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Ohms 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
• Common resistors used in electrical circuits
  range from a few W to MW (106W).
• If R is constant doesnt 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
  Ohms Law

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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.
• 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.

Fluid flow
Liquid flow
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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
r
Area A
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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.

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Temperature Dependence and Superconductivity
The resistivity of most materials depends on the
temperature. For most metals, resistivity
increases linearly with temperature over a small
range This temperature dependence to
resistivity can be exploited to build a
thermometer Measure the Voltage required to
maintain a fixed current in the resistor. Changes
in V measure changes in the temperature of the
medium surrounding the resistor. Some materials
(Pb, Nb, Nb3Sn, YBa2Cu3O7) when very cold (3 to
20 K), have a resistivity which abruptly drops to
zero. Such materials are then superconductors.
Highest temperature superconductor is T ? 100K
-173C
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Sample Resistivity values
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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

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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 V/R
Current is the same everywhere. Voltage varies
from point to point around loop.
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An Incandescent Light Bulb is a Resistor (but R
depends on Temperature T of filament, and T
depends on current I).
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Quiz 1 Which Circuit will light the bulb?

• A) B) C)

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Power in Electric Circuits
Recall that resistance is like an internal
friction - energy is dissipated. The amount
of energy dissipated when a charge DQ flows down
a voltage drop V in a time Dt is the power P P
?U/? t (DQV/Dt) IV SI unit watt, W
AmpVolt C V/s J/s For a resistor, PIV can
be rewritten with Ohms Law VIR, 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.
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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 0.10 per KW?hr
• My typical household uses 1KW of power, on
  average.
• There are 8800 hours in a year
• In one year, each household consumes 8800
  KW?hr, or 3.2 ? 1010 J at a cost of (8800
  KWhr)(0.10) 880.

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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.
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Circuits, Batteries, EMF

• Batteries and Electromotive Force (emf)
• Any device which increases the potential energy
  of charges which flow through it is called a
  source of emf.
• The emf is measured in volts and often written
  as e.
• The emf may originate from a chemical reaction as
  in a battery or from mechanical motion such
  as in a generator.

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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).
• VIR
• 5 Watt bulb with 3 V battery
• P V2 / R
• R V2/P (3V) 2/(5 AV)
• R 1.8 V/A 1.8 W.

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Direct Current (DC) Circuits - MORE
Includes batteries, resistors, capacitors
Kirchoffs Rules - conservation of charge
(Laws) follow from (junction rule, valid at
any junction) - conservation of
energy (loop rule, valid for any loop)
With emf (?) constant current can be
maintained charge pump forces electrons to
move in a direction opposite to the electric
field SI unit for emf Volt (V) No
resistance connecting wires of the loop
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Kirchhoffs 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 Kirchhoffs 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.
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A real battery is not an ideal emf. A simple
circuit with a battery and resistor can be
graphically represented as r is known
as the internal resistance of the battery. The
voltage on the terminals of the battery is,
therefore, V ? - Ir and the current in the
circuit is
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Battery as emf in the DC Circuits

terminal at higher potential then -
terminal V?-Ir V terminal voltage r
internal resistance ? - equivalent to
open-circuit (I0) voltage
I
Potential increases by ? Potential decreases by
Ir
-
I
r
Terminal voltage
Emf
?I
Total power of emf
Power dissipated as joule heat in
Internal resistor
Load resistor
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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 Parallel Combination
Combination
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Resistors in Series
b
a

• 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

c
d

• By Energy conservation (electrostatic potential
  is a function only of position in the circuit)
  potential drop around the loop from a to d equals
  the potential gain from d to a
• (Va-Vb) (Vb - Vc) (Vc - Vd) - e 0 -
  e (Va - Vd).
• Ohms law
• (Va-Vb)IR1, (Vb - Vc)IR2, (Vc - Vd) IR3
• e (Va-Vd) IReq I(R1R2R3) Req R1R2R3

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Resistors in Parallel
a

• By the conservation of charge,
• The current I splits into three (non-equal)
  branches such that II1I2I3.
• By Energy conservation potential drop across each
  resistor is the same (wires assumed to have zero
  resistance).
• (Va-Vd) - e 0.

d

• Ohms law
• (Va-Vd)I1R1, (Va - Vd)I2R2, (Va - Vd) I3R3
• I1 e/R1, I2 e/R2, I3 e/R3
• e I Req (I1I2I3) Req
• e (e/R1 e/R2 e/R3) Req e Req (1/R1
  1/R2 1/R3)1/ Req (1/R1 1/R2 1/R3)

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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.

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What is the Current in each resistor?
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Use Kirkhoffs Loop and Junction Laws to solve
for Currents

• Junction Law I1 I2 I3
• What is the current in the 9.8 W resistor?

• Loop Rule
• Left loop I1 (3.9 W) I3 (1.2 W) I1 (9.8
  W) 12 V 0
• I1 (13.7 W) I3 (1.2 W) 12 V 0
• Right loop I2 (6.7 W) I3 (1.2 W) 0 9.0 V
  0
• Three unknowns, three equations system is
  solvable.
• Use Junction eqn to eliminate I3 in all other
  equations
• I1 (13.7 W) (I1 I2) (1.2 W) 12 V 0 ?
  I1 (14.9 W) I2(1.2 W) 12 V 0
• I2 (6.7 W) (I1 I2) (1.2 W) 9.0 V 0 ?
  I1 (1.2 W) I2 (7.9 W) 9.0 V 0
• Solve left loop for I2 I2 12 V - I1 (14.9
  W) / (1.2 W)
• Substitute into right loop
• I1 (1.2 W) 12 V - I1 (14.9 W) (7.9 W)
  /(1.2 W) 9.0 V 0

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Final step in substitution--solving

• I1 (1.2 W) 12 V - I1 (14.9 W) (7.9 W) /(1.2
  W) 9.0 V 0

Now substitute backwards to get I2 and I3 I2
12 V - I1 (14.9 W) / (1.2 W) I2 12 V
(0.722A) (14.9 W) / (1.2 W) -0.764 A I1 I2
I3 I3 (0.722A) -0.764 A I3 -0.042 A
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Equivalent Resistance The current I is the
same in both The current may be different in
resistors, so the voltage Vba must each
resistor, but the voltage satisfy
Vba is the same across each VbaVa-VbIR1IR2I(R
1R2) resistor and the total current
is conserved II1I2 ReqR1R2 1/Req
1/R11/R2
c
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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 Parallel
Combination Combination
C2 C2
C1 C2
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Parallel vs. Series Combination Parallel Ser
ies charge Q1 , Q2 charge on each is
Q total QQ1 Q2 total charge is Q
voltage on each is V voltage V1 V2 V
Q1C1V QC1V1 Q2C2V QC2V2
QCeffV Q Ceff V Ceff(V1V2) CeffC1C2
1/Ceff1/C11/C2
C1 C2
Q
-Q
Q
-Q
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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 I0 in
branch with capacitor under study state
condition.
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Capacitor Charging Lets assume that at time t0,
the capacitor is uncharged, and we close the
switch. We can show that the charge on the
capacitor at some later time t
is qqmax(1-e-t/RC) RC is known as the time
constant ?, and qmax is the maximum amount
of charge that the capacitor will
acquire qmaxCe
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Capacitor Discharging Consider this circuit with
the capacitor fully charged at time t0 It
can be shown that the charge
on the capacitor is
given by qqmaxe-t/RC
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Ammeters and Voltmeters
Just as a real battery is not a perfect EMF, Real
ammeters and voltmeters are not perfect
either. An Ammeter can be represented as a small
resistance, with a perfect voltmeter in
parallel A voltmeter can be represented as a
large resistance, with a perfect voltmeter in
parallel. What does large or small mean?
Than must be defined in relation to your
circuit. All Ammeters Voltmeters will distort
the circuit they are trying to measure.
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