Electric Circuits

1
Electric Circuits

• Now that we have the concept of voltage, we can
  use this concept to understand electric circuits.
• Just like we can use pipes to carry water, we can
  use wires to carry electricity. The flow of
  water through pipes is caused by pressure
  differences, and the flow is measured by volume
  of water per time.

2
Electric Circuits

• In electricity, the concept of voltage will be
  like pressure. Water flows from high pressure to
  low pressure (this is consistent with our
  previous analogy that Voltage is like height
  since DP rgh for fluids) electricity flows
  from high voltage to low voltage.
• But what flows in electricity? Charges!
• How do we measure this flow? By Current
• current I Dq / Dt
• UNITS Amp(ere) Coulomb / second

3
Voltage Sourcesbatteries and power supplies

• A battery or power supply supplies voltage. This
  is analogous to what a pump does in a water
  system.
• Question Does a water pump supply water? If
  you bought a water pump, and then plugged it in
  (without any other connections), would water come
  out of the pump?
• Question Does the battery or power supply
  actually supply the charges that will flow
  through the circuit?

4
Voltage Sourcesbatteries and power supplies

• Just like a water pump only pushes water (gives
  energy to the water by raising the pressure of
  the water), so the voltage source only pushes the
  charges (gives energy to the charges by raising
  the voltage of the charges).
• Just like a pump needs water coming into it in
  order to pump water out, so the voltage source
  needs charges coming into it (into the negative
  terminal) in order to pump them out (of the
  positive terminal).

5
Voltage Sourcesbatteries and power supplies

• Because of the pumping nature of voltage
  sources, we need to have a complete circuit
  before we have a current.
• If we have an air gap (or rubber gap) in the
  circuit, no current will flow - just like if we
  have a solid block (like a cap) in a water
  circuit, no water will flow.
• If the gap is small, and the voltage is high
  enough, the current will cross over the gap -
  somewhat like water, if the pressure is high
  enough, will break through a plug.

6
Circuit Elements

• In this first part of the course we will consider
  two of the common circuit elements
• resistor
• capacitor
• The resistor is an element that resists the
  flow of electricity.
• The capacitor is an element that stores charge
  for use later (like a water tower).

7
Resistance

• Current is somewhat like fluid flow. Recall that
  it took a pressure difference to make the fluid
  flow due to the viscosity of the fluid and the
  size (area and length) of the pipe. So to in
  electricity, it takes a voltage difference to
  make electric current flow due to the resistance
  in the circuit.

8
Resistance

• By experiment we find that if we increase the
  voltage, we increase the current V is
  proportional to I. The constant of
  proportionality we call the resistance, R
• V IR Ohms Law
• UNITS R V/I so Ohm Volt / Amp.
• The symbol for resistance is W (capital omega).

9
Resistance

• Just as with fluid flow, the amount of resistance
  does not depend on the voltage (pressure) or the
  current (volume flow). The formula VIR relates
  voltage to current. If you double the voltage,
  you will double the current, not change the
  resistance.
• As was the case in fluid flow, the amount of
  resistance depends on the materials and shapes of
  the wires.

10
Resistance

• The resistance depends on material and geometry
  (shape). For a wire, we have
• R r L / A
• where r is called the resistivity (in Ohm-m) and
  measures how hard it is for current to flow
  through the material, L is the length of the
  wire, and A is the cross-sectional area of the
  wire. The second lab experiment deals with Ohms
  Law and the above equation.

11
Electrical Power

• The electrical potential energy of a charge is
• PE qV .
• Power is the change in energy with respect to
  time Power DPE / Dt .
• Putting these two concepts together we have
• Power D(qV) / Dt V(Dq) / Dt IV.

12
Electrical Power

• Besides this basic equation for power
• P IV
• remember we also have Ohms Law
• V IR .
• Thus we can write the following equations for
  power P I2R V2/R IV .
• To see which one gives the most insight, we need
  to understand what is being held constant.

13
Example

• When using batteries, the battery keeps the
  voltage constant. Each D cell battery supplies
  1.5 volts, so four D cell batteries in series
  (one after the other) will supply a constant 6
  volts.
• When used with four D cell batteries, a light
  bulb is designed to use 5 Watts of power. What
  is the resistance of the light bulb?

14
Example

• We know V 6 volts, and P 5 Watts were
  looking for R.
• We have two equations
• P IV and V IR
• which together have 4 quantities
• P, I, V R..
• We know two of these (P V), so we should be
  able to solve for the other two (I R).

15
Example

• Using the power equation we can solve for I
• P IV, so 5 Watts I (6 volts), or
• I 5 Watts / 6 volts 0.833 amps.
• Now we can use Ohms Law to solve for R
• V IR, so
• R V/I 6 volts / 0.833 amps 7.2 W .

16
Example extended

• If we wanted a higher power light bulb, should we
  have a bigger resistance or a smaller resistance
  for the light bulb?
• We have two relations for power that involve
  resistance
• PIV VIR eliminating V gives P I2R and
• PIV IV/R eliminating I gives P V2 / R .
• In the first case, Power goes up as R goes up in
  the second case, Power goes down as R goes up.
• Which one do we use to answer the above question?

17
Example extended

• Answer In this case, the voltage is being held
  constant due to the nature of the batteries.
  This means that the current will change as we
  change the resistance. Thus, the P V2 / R
  would be the most straight-forward equation to
  use. This means that as R goes down, P goes up.
  (If we had used the P I2R formula, as R goes
  up, I would decrease so it would not be clear
  what happened to power.)
• The answer for more power, lower the
  resistance. This will allow more current to flow
  at the same voltage, and hence allow more power!

18
Hooking Resistors Together

• Instead of making and storing all sizes of
  resistors, we can make and store just certain
  values of resistors. When we need a non-standard
  size resistor, we can make it by hooking two or
  more standard size resistors together to make an
  effective resistor of the value we need.
• The symbol for a resistor is written

19
Two basic ways

• There are two basic ways of connecting two
  resistors series and parallel.
• In series, we connect resistors together like
  railroad cars
• - -
• high V low R1 R2

20
Series

• If we include a battery as the voltage source,
  the series circuit would look like this
• R1
• Vbat
• R2
• Note that there is only one way around the
  circuit, and you have to go through BOTH
  resistors in making the circuit - no choice!

21
Parallel

• In a parallel hook-up, there is a branch point
  that allows you to complete the circuit by going
  through either one resistor or the other you
  have a choice!
• High V R1 Low V
• R2

22
Parallel Circuit

• If we include a battery, the parallel circuit
  would look like this
• Vbat R1 R2 - -

23
Formula for Series

• To see how resistors combine to give an effective
  resistance when in series, we can look either at
• V IR,
• or at
• R rL/A .

R1
I
V1
R2

Vbat
V2
-
24
Formula for Series

• Using V IR, we see that in series the current
  must move through both resistors.
• (Think of water flowing down two water falls in
  series.) Thus Itotal I1 I2 .
• Also, the voltage drop across the two resistors
  add to give the total voltage drop
• (The total height that the water fell is the
  addition of the two heights of the falls.)
• Vtotal (V1 V2). Thus, Reff Vtotal /
  Itotal
• (V1 V2)/Itotal V1/I1 V2/I2 R1 R2.

25
Formula for Series

• Using R rL/A , we see that we have to go over
  both lengths, so the lengths should add. The
  lengths are in the numerator, and so the values
  should add.
• This is just like in R V/I (from V IR) where
  the Vs are in the numerator and so add!

26
Formula for Parallel Resistors

• The result for the effective resistance for a
  parallel connection is different, but we can
  start from the same two places
• (Think of water in a river that splits with some
  water flowing over one fall and the rest falling
  over the other but all the water ending up
  joining back together again.) VIR, or R
  rL/A .

Itotal

Vbat
R1
R2
I1
I2
-
27
Formula for Parallel Resistors

• VIR, or R rL/A
• For parallel, both resistors are across the same
  voltage, so Vtotal V1 V2 . The current can go
  through either resistor, so Itotal (I1 I2
  ) .
• Since the Is are in the denominator, we have
• R Vtotal/Itotal Vtotal/(I1I2) or
• 1/Reff (I1I2)/Vtotal I1/V1 I2/V2
  1/R1 1/R2.

28
Formula for Parallel Resistors

• If we start from R rL/A , we can see that
  parallel resistors are equivalent to one resistor
  with more Area. But A is in the denominator
  (just like the current I was in the previous
  slide), so we need to add the inverses
• 1/Reff 1/R1 1/R2 .

29
Review

• Resistors V IR
• Power IV R rL/A
• Series Reff R1 R2
• Parallel 1/Reff 1/R1 1/R2
• series gives largest Reff , parallel gives
  smallest Reff .

30
Computer Homework

• The Computer Homework, Vol 3, 6, gives both an
  introduction and problems dealing with resistors.
  (For PHYS 202 you only need to do the first 5
  questions.)

31
Capacitance

• A water tower holds water. A capacitor holds
  charge.
• The pressure at the base of the water tower
  depends on the height (and hence the amount) of
  the water. The voltage across a capacitor
  depends on the amount of charge held by the
  capacitor.

32
Capacitance

• We define capacitance as the amount of charge
  stored per volt C Qstored / DV.
• UNITS Farad Coulomb / Volt
• Just as the capacity of a water tower depends on
  the size and shape, so the capacitance of a
  capacitor depends on its size and shape. Just as
  a big water tower can contain more water per foot
  (or per unit pressure), so a big capacitor can
  store more charge per volt.

33
Capacitance

• While we normally define the capacity of a water
  tank by the TOTAL AMOUNT of water it can hold, we
  define the capacitance of an electric capacitor
  as the AMOUNT OF CHARGE PER VOLT instead.
• There is a TOTAL AMOUNT of charge a capacitor can
  hold, and this corresponds to a MAXIMUM VOLTAGE
  that can be placed across the capacitor. Each
  capacitor DOES HAVE A MAXIMUM VOLTAGE.

34
Capacitance

• What happens when a water tower is over-filled?
  It can break due to the pressure of the water
  pushing on the walls.
• What happens when an electric capacitor is
  over-filled or equivalently a higher voltage is
  placed across the capacitor than the listed
  maximum voltage? It will break by having the
  charge escape. This escaping charge is like
  lightning - a spark that usually destroys the
  capacitor.

35
Capacitors

• As we stated before, the capacitance of a
  capacitor depends on its size and shape.
  Basically a capacitor consists of two separated
  (at least electrically separated) conductors
  (usually pieces of metal) so that we can pull
  charge from one and deposit it on the other.
• In the next slide we look at a common type of
  capacitor, the parallel plate capacitor where the
  two conductors are plates that are aligned
  parallel to each other each of area, A
  separated by a distance,d and containing
  anon-conductingmaterial betweenthe plates.

Top plate
A
d
Material between plates
Bottom plate
36
Parallel Plate Capacitor

• For a parallel plate capacitor, we can pull
  charge from one plate (leaving a -Q on that
  plate) and deposit it on the other plate (leaving
  a Q on that plate). Because of the charge
  separation, we have a voltage difference between
  the plates, DV. The harder we pull (the more
  voltage across the two plates), the more charge
  we pull C Q /DV.
• Note that C is NOT CHANGED by either Q or DV C
  relates Q and DV! The same applied to
  resistance the resistance did not depend on
  the current and voltage the resistance related
  the two.

Top plate
Q
A
d
DV
Material between plates
Bottom plate
-Q
37
V or DV ?

• When we deal with height, h, we usually refer to
  the change in height, Dh, between the base and
  the top. Sometimes we do refer to the height as
  measured from some reference point. It is
  usually clear from the context whether h refers
  to an actual h or a Dh.
• With voltage, the same thing applies. We often
  just use V to really mean DV. You should be able
  to determine from the context whether we really
  mean V or DV when we say V.

38
Parallel Plate Capacitor

• For this parallel plate capacitor, the
  capacitance is related to charge and voltage (C
  Q/V), but the actual capacitance depends on the
  size and shape
• Cparallel plate K A / (4 p k d)
• where K (called dielectric constant) depends on
  the material between the plates, A is the area of
  each plate, d is the distance between the plates,
  and k is Coulombs constant (9 x 109 Nt-m2 /
  Coul2).

Top plate
Q
A
d
DV
Material between plates
Bottom plate
-Q
39
Example Parallel Plate Capacitor

• Consider a parallel plate capacitor made from two
  plates each 5 cm x 5 cm separated by 2 mm with
  vacuum in between. What is the capacitance of
  this capacitor?
• Further, if a power supply puts 20 volts across
  this capacitor, what is the amount of charged
  stored by this capacitor?

40
Example Parallel Plate Capacitor

• The capacitance depends on K, A, k and d
• Cparallel plate K A / (4 p k d)
• where K 1 for vacuum, A 5 cm x 5 cm 25 cm2
  25 x 10-4 m2, d 2 mm 2 x 10-3 m, and k 9
  x 109 Nt-m2/Coul2 , so C
• (1) (25 x 10-4 m2) / 4 3.14 9 x 109
  Nt-m2/Coul2 2 x 10-3 m 1.10 x 10-11 F
  11 pF .

41
Other types of capacitors

• Note We can have other shapes for capacitors.
  These other shapes will have formulas for them
  that differ from the above formula for parallel
  plates. These formulas will also show that the
  capacitance depends on the materials and shape of
  the capacitor.

42
Example (cont.)

• We can see from the previous example thata Farad
  is a huge capacitance!
• If we have a DV 20 volts, then to calculate the
  charge, Q, we can use C Q/V to get
• Q CV 11 x 10-12 F 20 volts
• 2.2 x 10-10 Coul 0.22 nCoul 220 pCoul.
• Remember that we often drop the D in front of
  the V since we often are concerned by the change
  in voltage rather than the absolute value of the
  voltage - just as we do when we talk about height!

43
Capacitance

• Note that if we doubled the voltage, we would not
  do anything to the capacitance. Instead, we
  would double the charge stored on the capacitor.
• However, if we try to overfill the capacitor by
  placing too much voltage across it, the positive
  and negative plates will attract each other so
  strongly that they will spark across the gap and
  destroy the capacitor. Thus capacitors have a
  maximum voltage!

44
Energy Storage

• If a capacitor stores charge and carries voltage,
  it also stores the energy it took to separate the
  charge. The formula for this is
• Estored (1/2)QV (1/2)CV2 ,
• where in the second equation we have used the
  relation C Q/V .

45
Energy Storage

• Note that previously we had
• PE qV ,
• and now for a capacitor we have
• E (1/2)QV .
• Why the 1/2 factor for a capacitor?

46
Energy Storage

• The reason is that in charging a capacitor, the
  first bit of charge is transferred while there is
  very little voltage on the capacitor (recall that
  the charge separation creates the voltage!).
  Only the last bit of charge is moved across the
  full voltage. Thus, on average, the full charge
  moves across only half the voltage!

47
Hooking Capacitors Together

• Instead of making and storing all sizes of
  capacitors, we can make and store just certain
  values of capacitors. When we need a
  non-standard size capacitor, we can make it by
  hooking two or more standard size capacitors
  together to make an effective capacitor of the
  value we need. (Similar to what we saw with
  resistors.)

48
Two basic ways

• Just as with resistors, there are two basic ways
  of connecting two capacitors series and
  parallel. In series, we connect capacitors
  together like railroad cars using parallel plate
  capacitors it would look like this
• - -
• high V low V
• C1 C2

49
Series

• If we include a battery as the voltage source,
  the series circuit would look like this
• C1
• Vbat
• C2
• Note that there is only one way around the
  circuit, and you have to jump BOTH capacitors in
  making the circuit - no choice!

-

-
50
Parallel

• In a parallel hook-up, there is a branch point
  that allows you to complete the circuit by
  jumping over either one capacitor or the other
  you have a choice!
• High V C1 Low V
• C2

-

-
51
Parallel Circuit

• If we include a battery, the parallel circuit
  would look like this
• Vbat C1 C2

52
Formula for Series

• To see how capacitors combine to give an
  effective capacitance when in series, we can look
  either at C Q/V, or at
• Cparallel plate KA / 4pkd .

53
Formula for Series

• Using C Q/V, we see that in series the charge
  moved from capacitor 2s negative plate must be
  moved through the battery to capacitor 1s
  positive plate.
• C1
• Q
• Vbat C2
• - -Q
• ( ? Qtotal)

54
Formula for Series

• But the positive charge on the left plate of C1
  will attract a negative charge on the right
  plate, and the negative charge on the bottom
  plate of C2 will attract a positive charge on the
  top plate - just what is needed to give the
  negative charge on the right plate of C1. Thus
  Qtotal Q1 Q2 .
• C1 (Q1 ? )
• Q1 -Q 1 Q2
• Vbat C2
• - -Q2
• ( ? Qtotal)

55
Formula for Series

• Also, the voltage drop across the two capacitors
  add to give the total voltage drop
• Vtotal (V1 V2).
• Thus, Ceff Qtotal / Vtotal Qtotal /
  (V1 V2), or (with Qtotal Q1 Q2)
• 1/Ceff (V1 V2) / Qtotal V1/Q1
  V2/Q2 1/C1 1/C2 1/Ceffective .
• Note this is the opposite of resistors when
  connected in series! Recall that R V/I where V
  is in the numerator but with capacitors C Q/V
  where V is in the denominator!

56
Formula for Series

• Using Cparallel plate KA / 4pkd , we see that
  we have to go over both distances, so the
  distances should add. But the distances are in
  the denominator, and so the inverses should add.
  This is just like in C Q/V where the Vs add
  and are in the denominator!

57
Formula for Parallel Capacitors

• The result for the effective capacitance for a
  parallel connection is different, but we can
  start from the same two places
• C Q/V, or Cparallel plate KA / 4pkd .

58
Parallel Circuit

• For parallel, both plates are across the same
  voltage, so Vtotal V1 V2 . The charge can
  accumulate on either plate, so Qtotal (Q1
  Q2).
• Since the Qs are in the numerator of C Q/V, we
  have
• Ceff C1 C2.
• Q1 Q2
• Vbat C1 -Q1 C2 -Q2
• Q1 ?
• ? Qtotal (Q1Q2) ? Q2

59
Formula for Parallel Capacitors

• If we use the parallel plate capacitor formula,
• Cparallel plate KA / 4pkd , we see that the
  areas add, and the areas are in the numerator,
  just as the Qs were in the numerator in the C
  Q/V definition.

60
Review of Formulas

• For capacitors in SERIES we have
• 1/Ceff 1/C1 1/C2 .
• For capacitors in PARALLEL we have
• Ceff C1 C2 .
• Note that adding in series gives Ceff being
  smaller than the smallest, while adding in
  parallel gives Ceff being larger than the largest!

61
Review

• Capacitors C Q/V
• PE ½CV2 C// KA/4pkd
• Series 1/Ceff 1/C1 1/C2
• Parallel Ceff C1 C2
• series gives smallest Ceff , parallel gives
  largest Ceff .
• Resistors V IR
• Power IV R rL/A
• Series Reff R1 R2
• Parallel 1/Reff 1/R1 1/R2
• series gives largest Reff , parallel gives
  smallest Reff .

62
Computer Homework

• The Computer Homework, Vol 3, 5, gives both an
  introduction and problems dealing with
  capacitors. (For PHYS 202 you only need to do
  the first four questions.)