Humans are clever!

1
Humans are clever!

• With the magnetic force only pushing
  perpendicular to the motion, it would seem that
  we could not use this force to increase the speed
  of particles and hence generate energy.
• However, humans being clever, consider the
  following slides.

2
Generating Currents

• Consider the following circuit a bar moves on
  two rails that are connected at one end. The
  whole setup has a magnetic field that goes
  through it.
• B Ä Ä
• v
• Ä Ä

3
Generating Currents

• If the bar is moving to the right, and the bar
  contains electrons that are free to move (as in
  metal), then the electrons are also moving to the
  right. There is a magnetic force on the
  electrons pushing them down with magnitude
  Fmagnetic q v B.
• B Ä Ä
• v
• Fmag on e
• Ä Ä

4
Generating Currents

• Thus, negative electrons should pile up at the
  bottom of the moving bar, leaving a net positive
  charge at the top of the bar. But this should
  act just like a battery!
• B Ä Ä
• v
• Fmag on e
• Ä - Ä

5
Generating Currents

• To find the voltage of this battery, we note
  that as the charges pile up at the ends of the
  rod, an Electric field will be set up. The
  electrons will continue to pile up until the
  Electric Force
• (Fel qE) balances the Magnetic Force.
• B Ä Ä
• Felec on e v
• Fmag on e
• Ä - Ä

6
Generating a Voltage

• We now have, at equilibrium SF 0, or
• Felec on e Fmag on e , or qE qvB , or
• E vB .
• We know that the electric field is related to
  voltage by E -DV / Ds , (here Ds L, the
  length of the bar). Thus we have for the
  voltage DV v B L .

7
Generating Power

• We have generated a voltage, and now to generate
  electric power (PIV) we need to have that
  voltage drive a current. Since we have completed
  the circuit by connecting the ends of the rails,
  we will have a complete circuit - and so we will
  get a current depending on the resistance in the
  rails (VIR).

8
Conservation of Energy

• We have made an electric generator that can
  generate electrical energy. But according to the
  Law of Conservation of Energy, we can only
  convert energy from one form into another. In
  the case of the electric generator, where does
  this energy come from?

9
Generating Currents

• Note that as electrons flow clockwise around the
  circuit, this acts the same as a current of
  positive charges going counterclockwise, as
  indicated on the diagram below. Note that a
  current flows up the bar.
• B Ä I Ä
• I Felec on e v
• Fmag on e I
• I Ä - Ä

10
Generating Currents

• Is there a magnetic force on this current due to
  its flowing through a magnetic field? YES!
• Note that the direction of the force on this
  current is to the left. This will act to slow
  the bar down. In effect, this apparatus converts
  the kinetic energy of the bar into electric
  energy!
• B Ä I Ä
• I Felec on e v
• Fmag on e I Fmag on I
• I Ä - Ä

11
Generalizing

• We have from the previous apparatus
• DV v B L .
• We note that v Dw/Dt where w is the width of
  the circuit (distance from end of rails to bar).
  Can we take the D /Dt and apply it to all the
  variables DV D(B L w) / Dt ?
• The answer, based on experiment, is YES!

12
Faradays Law

• We also note that wL A (area of circuit). We
  can also have N number of loops, so we finally
  get DV D(N B A) / Dt . This is called
  Faradays Law.
• When we consider direction as well, we see that
  the magnetic field, B, has to cut through the
  area, A. If we assign a direction to A that is
  perpendicular to the surface, we get an even more
  general form
• DV D (N B A cos(qBA) / Dt .

13
Lenzs Law

• DV D (N B A cos(qBA) / Dt
• The above formula is for determining the amount
  of voltage generated. But what is the direction
  of that voltage (what direction will it try to
  drive a current)?
• The answer is Lenzs Law the direction of the
  induced voltage will tend to induce a current to
  oppose the change in magnetic field through the
  area.

14
Lenzs Law

• Well go over several cases in class. Well also
  have a lab later in the semester to play with
  this.
• The Computer Homework assignment, Vol. 4 3,
  deals with Lenzs Law and will give you practice
  with this as well.

15
Transformers and Inductors

• DV D (N B A cos(qBA) / Dt .
• We have already seen how changing the area of a
  circuit in a magnetic field generates a voltage.
• We could also change the magnetic field strength
  through the circuit to generate a voltage - this
  is the basis of an inductor and a transformer.
• Well consider both a little later after we look
  at AC voltages.

16
Electric Generators

• Finally, we could change the direction of the
  area in relation to the field - this is the basis
  for the most common kind of generator. This
  generator looks just like the electric motor,
  except we put in rotational motion and get
  current instead of putting in current and getting
  rotational motion!

17
Electric Generator

• N S
• crank (turn at frequency, f)

18
Electric Generators

• DV D (N B A cos(qBA) / Dt .
• When we change the angle, qBA, with respect to
  time (qBAwt) , the calculus gives us the
  following relation DV N B A w sin(qBA) , or
  VAC Vo sin(wt) where VoNBAw, and where w
  DqBA/Dt 2pf.
• This kind of voltage is an alternating voltage
  (AC voltage) since the sine function alternates
  between positive and negative.

19
AC Circuits

• VAC Vo sin(wt)
• For this kind of AC voltage, we can determine the
  amplitude of the voltage (Vo NBAw) . But since
  the average of sine is zero, how do we treat the
  average?
• What is usually important is the power delivered
  by the electric circuit. From PIV we see that
  both the current and the voltage are important.

20
AC Circuits

• From Ohms Law, we have V IR, where R is a
  constant that depends on the material and
  geometry of the materials used to conduct the
  current. Thus, I V/R (NBAw/R) sin(wt) Io
  sin(wt) , where
• Io NBAw/R Vo/R . The alternating voltage
  creates an alternating current!
• From this we see that the electric power is
• P I V IoVo sin2(wt) .

21
AC Power

• P I V IoVo sin2(wt)
• Note that the Power involves the square of the
  sine function, and so the Power oscillates but is
  always positive.
• But what we are usually interested in is the
  average power. From the calculus, we find that
  the average of sin2(q) 1/2. Thus
• Pavg (1/2)IoVo .

22
RMS Voltage and Current

• In order to work with AC circuits just as we did
  with DC circuits, we create a voltage and current
  called rms (root mean square).
• Vrms Vo (1/2)1/2 and Irms Io (1/2)1/2
• so that we have
• Pavg Irms Vrms and Vrms Irms R .
• Note that the power formula and Ohms Law are the
  same for DC and for AC-rms, but NOT for
  instantaneous AC.

23
Transformers

• Transformers work this way An AC voltage
• VAC-1 V1 sin(wt) , is used to generate a
  current IAC-1 I1sin(wt) in coil 1 which
  generates an oscillating magnetic field in coil
  1 since B is proportional to I BAC-1
  B1sin(wt) a second coil (2) is inside coil 1
  this 2 coil then, by Faradays Law, DV D (N
  B A cos(qBA) / Dt
• has a voltage, VAC-2, induced in it. By
  adjusting the number of loops in both coils, the
  induced (AC) voltage in 2 can be different than
  that in coil 1. We can adjust (or transform)
  the voltage up or down!

24
Transformers

• For safety, we like to have rather low voltages
  in the house.
• For economy, since Plost I2R, we like to have
  low currents (which means high voltages) in our
  transmission lines.
• We can have both if we use transformers located
  in our neighborhoods!

25
Inductors

• A coil of wire can create a magnetic field if a
  current is run through it. If that current
  changes (as in the AC case), the magnetic field
  created by the coil will change. Will this
  changing magnetic field due to the changing
  current through the coil cause a voltage to be
  created across the coil? YES!
• This is called self-inductance and is the basis
  behind the circuit element called the inductor.

26
Inductors

• Since the voltage created depends in this case on
  the changing magnetic field,
• DV D (N B A cos(qBA) / Dt .
• and the field depends on the changing current,
• B (mo/4p) I DL sin(qIr) / r2
• we have Vinductor -L DI / Dt
• where the L (called the inductance) depends on
  the shape and material
• (just like capacitance and resistance).

27
Inductors

• Vinductor -L DI / Dt
• Here the minus sign means that when the current
  is increasing, the voltage across the inductor
  will tend to oppose the increase, and it also
  means when the current is decreasing, the voltage
  across the inductor will tend to oppose the
  decrease.

28
Units Henry

• From Vinductor -L dI /dt
• L has units of Volt / Amp/sec which is called
  a Henry
• 1 Henry Volt-sec / Amp .
• For a solenoid L mpN2R2 / Length
• (A Henry is a rather large amount of inductance.)

29
Energy Stored in an Inductor

• We start from the definition of voltage V
  PE/q (or PE qV). But since the voltage
  across an inductor is related to the current
  change, we might express q in terms of I
• I dq/dt, or dq I dt. Therefore, we have
• Estored S qi Vi ? V dq ? V I dt and
  now we use VL L dI/dt to get
• Estored ? (L dI/dt) I dt ? L I dI
  (1/2)LI2.

30
Review of Energy in Circuits

• There is energy stored in a capacitor (that has
  Electric Field) Estored (1/2)CV2 .
• Recall that V is related to E (Electric Field).
• There is energy stored in an inductor (that has
  Magnetic Field) Estored (1/2)LI2 .
• Recall the I is related to B (Magnetic Field).
• There is power dissipated (as heat) in a
  resistor Plost RI2 .

31
Review of Circuit Elements

• Resistor VR R I where I Dq/Dt
• Capacitor VC (1/C)q (from C q/V)
• Inductor VL -L DI/Dt
• We can make an analogy with mechanics
• q is like x V is like F
• t is like t L is like m
• I Dq/Dt is like v Dx/Dt C is like 1/k
  (spring)
• DI/Dt is like a Dv/Dt R is like air
  resistance.

32
RL Circuit - DC

• What happens when we have a resistor in series
  with an inductor in a circuit with a battery?
• From Conservation of Energy, the voltage changes
  around the circuit must equal zero
• Vbattery Vresistor Vinductor 0 , or
• Vbattery IR LdI/dt .
• If we replace I with v, and dI/dt with a, and R
  with b (air resistance), and L with m, and
  identify Vbattery constant with the constant
  force of gravity, mg, we have an equivalent
  equation
• mg bv ma, or mg bv ma (with down
  being the positive direction for a falling
  object).

33
RL Circuit - DC

• From the mechanical analogy, this should be like
  having a mass with air resistance. If we have a
  constant force (like gravity), the object will
  accelerate up to a terminal speed (due to force
  of air resistance increasing up to the point
  where it balances the gravity).
• SFma ? -bv mg ma, or
• m dv/dt bv mg

34
Mechanical Analogy mass falling with air
resistance

35
RL Circuit - DC (cont.)

• If we connect the resistor and the inductor to a
  battery and then turn the switch on, from the
  mechanical analogy we would expect the current
  (which is like velocity) to begin to increase
  until it reaches a constant amount.
• From conservation of energy
• Vbattery Vresistor Vinductor where
• Vbattery constant VR IR, and VL LdI/dt .

36
LR Circuit - qualitative look

• From the circuit point of view, initially we have
  zero current so there is no VR (voltage drop
  across the resistor). Thus the full voltage of
  the battery is trying to change the current,
  hence VL Vbattery, and so dI/dt Vbattery /L.
• However, as the current increases, there is more
  voltage drop across the resistor, VR, which
  reduces the voltage across the inductor (VL
  Vbattery - VR), and hence reduces the rate of
  change of the current!

37
RL Circuit - DC I(t) versus t

38
Review of Circuit Elements

• Resistor VR R I where I Dq/Dt
• Capacitor VC (1/C)q (from C q/V)
• Inductor VL -L DI/Dt
• We can make an analogy with mechanics
• q is like x V is like F
• t is like t L is like m
• I Dq/Dt is like v Dx/Dt C is like 1/k
  (spring)
• DI/Dt is like a Dv/Dt R is like air
  resistance.

39
Inductive Reactance - AC

• Since VL -L ?I/?t, for an AC current we will
  have a voltage induced that will oppose the
  changing current. This opposition will tend to
  limit the current in the circuit and behave in
  some sense like a resistance. We call this
  action Reactance.
• For an inductor, since VL -L ?I/?t, VL will be
  big if L is big and/or if ? is big (causing ?I/?t
  to be big). A more detailed calculation for
  reactance (or resistance to an AC circuit) gives
  XL ?L and is used in the Ohms Law-like
  relation VL XLI (the bigger XL, the smaller
  the I or the bigger the VL)

40
Capacitive Reactance - AC

• There is a similar effect for a capacitor in an
  AC circuit. For a capacitor, since VC q/C, VC
  will be small if C is big and/or if ? is big
  (causing little q to accumulate in the short
  time).
• A more detailed calculation for reactance gives
  XC 1/(?C), and it is used in the Ohms Law-like
  relation VC XCI
• (the bigger XC, the smaller the I or the bigger
  the VC)

41
AC Circuits

• A resistor obviously limits the current in a
  circuit. But, as we just saw, a capacitor and an
  inductor also limit the current in an AC circuit.
  However, the reactances do not just add
  together. Using the fundamental relations and
  the calculus, we come up with the concept of
  impedance, Z V IZ where Z takes into
  account all three reactances XRR, XL?L and
  XC 1/?C
• Z R2 (wL - 1/wC)21/2.
• Power, however, is still Pavg I2R (not
  PI2Z).

42
Oscillations

• Newtons Second Law SF ma can be written as
  SF - ma 0 .
• This is like Conservation of Energy SV 0 .
• If we put an inductor with a capacitor with an AC
  voltage, we have the analogy with a mass
  connected to a spring that has an oscillating
  applied force.
• In each of these cases, we get resonance.
• Well demonstrate this in class with a mass and
  spring. This is the basis of tuning a radio!

43
AC Circuits

• V IZ where Z R2 (wL - 1/wC)21/2 .
• Note that when (wL - 1/wC) 0, Z is smallest and
  so I is biggest! This is the condition for
  resonance. Thus when w 1/LC1/2, we have
  resonance. This is equivalent to the resonance of
  a spring when w k/m1/2 .
• Computer Homework Vol. 4, 4, gives practice with
  problems involving inductance (L) and impedance
  (Z).