Tuesday April 19, 2011 1PHYS 1444-002, Dr. Andrew BrandtPHYS 1444 – Section 02Lecture #18Tuesday April 19, 2011Dr. Andrew BrandtChapter 29• Lenz Law• Generator• TransformerTuesday April 19, 2011 PHYS 1444-002, Dr. Andrew Brandt2Electromagnetic Induction• Well known that a current produces a magnetic field, so by symmetry might expect that a magnetic field can produce a current (or voltage)• It was found that relative motion of a wire coil and a magnet induces a current (and consequently an emf) in the wire.Tuesday April 19, 2011 3Magnetic Flux• So what do you think the induced emf is proportional to?– The rate of changes of the magnetic field?• the higher the change the larger the induced EMF? – Close! It actually depends on the rate of change of the magnetic flux, B.– Magnetic flux is similar to electric flux–cosBB A BA B ABB dA2Tm211Wb T m• is the angle between B and the area vector A whose direction is perpendicular to the face of the loop based on the right-hand rule–What kind of quantity is the magnetic flux?•Scalar. Unit?• or weber•If the area of the loop is not simple or B is not uniform, the magnetic flux can be written asTuesday April 19, 2011 PHYS 1444-002, Dr. Andrew Brandt4Faraday’s Law of Induction• In terms of magnetic flux, we can formulate Faraday’s findings– The emf induced in a circuit is equal to the rate of change of magnetic flux through the circuitFaraday’s Law of InductionBdNdt• For a single loop of wire N=1, for closely wrapped loops, N is the number of loops• The negative sign has to do with the direction of the induced emf (Lenz’s Law)Tuesday April 19, 2011 PHYS 1444-002, Dr. Andrew Brandt5Lenz’s Law• It is experimentally found that– An induced emf gives rise to a current whose magnetic field opposes the original change in flux This is known as Lenz’s Law– We can use Lenz’s law to explain the following cases in the figures• When the magnet is moving into the coil– Since the external flux increases, the field inside the coil takes the opposite direction to minimize the change and causes the current to flow clockwise• When the magnet is moving out– Since the external flux decreases, the field inside the coil takes the opposite direction to compensate the loss, causing the current to flow counter-clockwise• Which law is Lenz’s law result of?– Energy conservation. Why? (no free lunch)Tuesday April 19, 2011 PHYS 1444-002, Dr. Andrew Brandt6Induction of EMF• How can we induce an emf?• Let’s look at the formula for magnetic flux•• What do you see? What are the things that can change with time to result in change of magnetic flux?– Magnetic field– The area of the loopB– The angle between the field and the area vectorB dAcosB dATuesday April 19, 2011 PHYS 1444-002, Dr. Andrew Brandt7Example 29 – 2 Pulling a coil from a magnetic field. A square coil of wire with side 5.00cm contains 100 loops and is positioned perpendicular to a uniform 0.600-T magnetic field. It is quickly and uniformly pulled from the field (moving perpendicular to B) to a region where B drops abruptly to zero. At t=0, the right edge of the coil is at the edge of the field. It takes 0.100s for the whole coil to reach the field-freeWhat should be computed first?The flux at t=0 isThe change of flux isregion. Find (a) the rate of change in flux through the coil, (b) the emf and current induced, and (c) how much energy is dissipated in the coil if its resistance is 100 . (d) what was the average force required? The initial flux at t=0. BThus the rate of change of the flux is330 1.50 10 1.50 10BWb WbBtBABA220.600 5 10Tm31.50 10 Wb0.10 0s21.50 10 Wb s31.50 10 WbTuesday April 19, 2011 PHYS 1444-002, Dr. Andrew Brandt8Example 29 – 2, cnt’d Thus the total emf induced in this period is Which direction would the induced current flow?The total energy dissipated isForce for each coil isEIThe induced current in this period is Clockwise to compensate for the loss of flux through coilFForce for N coil isFFBdNdt2100 1.50 10 1.5Wb s VR21.51.50 10 15.0100VA mAPt2I Rt2231.50 10 100 0.100 2.25 10A s JIl BNIl BNIlB22100 1.50 10 4 5 10 0.600 0.045A T NTuesday April 19, 2011 PHYS 1444-002, Dr. Andrew Brandt9EMF Induced on a Moving Conductor• Another way of inducing emf is using a U shaped conductor with a movable rod resting on it.• As the rod moves at a speed v, it travels vdt in time dt, changing the area of the loop by dA=lvdt.BddtBdAdtdtBlvBlvdt• Using Faraday’s law, the induced emf for this loop is–This equation is valid as long as B, l and v are perpendicular to each other. •An emf induced on a conductor moving in a magnetic field is called a motional emfTuesday April 19, 2011 PHYS 1444-002, Dr. Andrew Brandt10Electric Generator (Dynamo)• An electric generator transforms mechanical energy into electrical energy• It consists of many coils of wires wound on an armature that can be rotated in a magnetic field• An emf is induced in the rotating coil• Electric current is the output of a generator• Which direction does the output current flow when the armature rotates counterclockwise?– Initially the current flows as shown in figure to reduce flux through the loop– After half a revolution, the current flow is reversed• Thus a generator produces alternating currentTuesday April 19, 2011 PHYS 1444-002, Dr. Andrew Brandt11How does an Electric Generator work?• Let’s assume the loop is rotating in a uniform B field with constant angular velocity The induced emf is•BddtdB dAdtcosdBAdtcosdBA tdtsinBA tBdNdtsinNBA t0sin t• What is the variable that changes above?–The angle . What is d /dt?•The angular speed .–So t–If we choose 0=0, we obtain––If the coil contains N loops:–What is the shape of the output?•Sinusoidal w/ amplitude 0=NBA• US AC frequency is 60Hz. Europe uses 50HzTuesday April 19, 2011 PHYS 1444-002, Dr. Andrew Brandt12Example 29 – 5 An AC generator. The armature of a 60-Hz AC generator rotates in a 0.15-T magnetic field. If the area of the coil is 2.0x10-2m2, how many loops must the coil contain if the peak output is to be 0=170V?The maximum emf of a generator is Solving for NSince2 f0NBA0NBAWe obtainN02 BAf2 2 11701502 0.15 2.0 10 60VturnsT m sTuesday April 19, 2011 PHYS 1444-002, Dr. Andrew Brandt13A DC Generator• A DC generator is almost the same as an ac generator except the slip rings are replaced by
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