Physics 122B Electricity and Magnetism

1. Physics 122B Electricity and Magnetism Lecture 22 (Knight: 33.5 to 33.7) Faraday’s Law Martin Savage

2. Lecture 21 Announcements • Lecture HW has been posted and is due on Wednesday at 10 PM. • Next Friday we will have Exam 3 in this room. It will consist of multiple-choice questions on the Laboratory (25 pts) and Lecture (35 pts). Bring a Scatron sheet, a double-sided page of notes, and a calculator with good batteries. Physics 122B - Lecture 22

3. Question What is the ranking of the forces in the figure? (a) F1=F2=F3=F4; (b) F1<F2=F3>F4; (c) F1=F3<F2=F4; (d) F1=F4<F2=F3; (e) F1<F2<F3=F4; Physics 122B - Lecture 22

4. Magnetic Flux • The number of arrows passing through the loop depends on two factors: • (1) The density of arrows, which is proportional to B • The effective areaAeff = A cos q of the loop • We use these ideas to define the magnetic flux: Physics 122B - Lecture 22

5. Area Vector Define the area vector A of a loop such that it has the loop area as its magnitude and is perpendicular to the plane of the loop. If a current is present, the area vector points in the direction given by the thumb of the right hand when the fingers curl in the direction of current flow. If the area is part of a closed surface, the area vector points outside the enclosed volume. With this definition: Physics 122B - Lecture 22

6. Example: A Circular Loop Rotating in a Magnetic Field The figure shows a 10 cm diameter loop rotating in a uniform 0.050 T magnetic field. What is the magnitude of the flux through the loop when the angle is q=00, 300, 600, and 900? Physics 122B - Lecture 22

7. Magnetic Fluxin a Nonuniform Field So far, we have assumed that the loop is in a uniform field. What if that is not the case? The solution is to break up the area into infinitesimal pieces, each so small that the field within it is essentially constant. Then: Physics 122B - Lecture 22

8. Example: Magnetic Flux froma Long Straight Wire The near edge of a 1.0 cm x 4.0 cm rectangular loop is 1.0 cm from a long straight wire that carries a current of 1.0 A, as shown in the figure. What is the magnetic flux through the loop? Physics 122B - Lecture 22

9. Lenz’s Law (1) Heinrich Friedrich Emil Lenz (1804-1865) In 1834, Heinrich Lenz announced a rule for determining the direction of an induced current, which has come to be known as Lenz’s Law. Here is the statement of Lenz’s Law: There is an induced current in a closed conducting loop if and only if the magnetic flux through the loop is changing. The direction of the induced current is such that the induced magnetic field opposes the change in the flux. Physics 122B - Lecture 22

10. Lenz’s Law (2) If the field of the bar magnet is already inthe loop and the bar magnet is removed, theinduced current is in the direction that triesto keep the field constant. If the loop is a superconductor, a persistentstanding current is induced in the loop, and thefield remains constant. Superconductingloop Physics 122B - Lecture 22

11. Six Induced Current Scenarios Physics 122B - Lecture 22

12. Example: Lenz’s Law 1 - + - + The switch in the circuit shown has been closed for a long time. What happens to the lower loop when the switch is opened? Physics 122B - Lecture 22

13. Example: Lenz’s Law 2 + - The figure shows two solenoids facing each other. When the switch for coil 1 is closed, does the current in coil 2 flow from right to left or from left to right? Physics 122B - Lecture 22

14. Example: A Rotating Loop A loop of wire is initially in the xy plane in a uniform magnetic field in the x direction. It is suddenly rotated 900 about the y axis, until it is in the yz plane. In what direction will be the induced current in the loop? Initially there is no flux through the coil. After rotation the coil will be threaded by magnetic flux in the x direction. The induced current in the coil will oppose this change by producing flux in the –x direction. Let your thumb point on the –x direction, and your fingers will curl clockwise. Therefore, the induced current will be clockwise, as shown in the figure. Physics 122B - Lecture 22

15. Faraday’s Law Consider the loop shown: This is Faraday’s Law. It can be stated as follows: An emf E is induced in a conducting loop if the magnetic flux Fm through the loop changes with time, so that E = |dFm/dt| for the loop. The emf will be in the direction that will drive the induced current to oppose the flux change, as given by Lenz’s Law. Physics 122B - Lecture 22

16. Example: Electromagnetic Induction in a Circular Loop The magnetic field shown in the figure decreases from 1.0 T to 0.4 T in 1.2 s. A 6.0 cm diameter loop with a resistance of 0.010 W is perpendicular to the field. What is the size and direction of the current induced in the loop? I The current direction is such as to reinforce the diminishing B field. Therefore, the current I will be clockwise. Physics 122B - Lecture 22

17. Example: Electromagnetic Induction in a Solenoid A 3.0 cm diameter loop with a resistance of 0.010 W is placed in the center of a solenoid. The solenoid is 4.0 cm in diameter, 20 cm long, and is wound with 1000 turns of square insulated wire. The current through the solenoid wire as a function of time is shown in (b). Find the induced current in the loop. Physics 122B - Lecture 22

18. What does Faraday’sLaw Tell Us? • Faraday’s Law tells us that all induced currents are the associated with a changing magnetic flux. There are two fundamentally different ways to change the magnetic flux through a loop: • The loop can move, change size, or rotate, creating motional emf; • The magnetic field can change in magnitude or direction. • We can write: new physics motional emf The second term says that an emf can be created simply by changing a magnetic field, even if nothing is moving. Physics 122B - Lecture 22

19. An Unanswered Question A very long solenoid with no field outside passes through a conducting loop. The current in the solenoid is increased so that the B field inside the solenoid increases. (B outside = 0). There is no B-field at the loop wire. Is a current induced in the loop? YES! Since the flux through the loop changes, an emf is induced in the loop, even though the field that produces the flux does not touch the loop. How can this happen? Faraday would say that when the number of lines of force in the solenoid increases, they must “come in” from infinity and must cut through the loop on their way in. Physics 122B - Lecture 22

20. Question A conducting loop is half way into a magnetic field. Suppose that the field begins to increase rapidly in strength. Which statement describes the behavior of the loop? • The loop is pushed upward, toward the top of the page; • The loop is pushed downward, toward the bottom of the page; • The loop is pushed to the left, into the magnetic field; • The loop is pushed to the right, out of the magnetic field; • The tension in the wire increases, but the wire does not move. Physics 122B - Lecture 22

21. Induced Fields and Electromagnetic Waves • There is still a puzzle piece missing.Faraday’s Law allows us to calculate an inducedcurrent, but what causes the current? Whatforce pushes the electrons around in the wire? If the wire is stationary, there can be no motional vxB magnetic force. Therefore, there must be an induced electric field. Thus, there are two ways to create an electric field: • A Coulomb electric field that is created by positive or negative charges; • A non-Coulomb electric field that is created by a changing magnetic field. Physics 122B - Lecture 22

22. Maxwell’s Theory Maxwell produced a mathematical formulation of Faraday’s lines of force picture. He reasoned from this that if a changing magnetic field produces an electric field, then a changing electric field should be equivalent to a current in producing a magnetic field. Otherwise, there is a paradox. An Amperian loop near a charging capacitor will predict a different magnetic field, depending on whether the surface enclosed by the loop passes throughthe current (a) or through thecapacitor gap (b). If the changing electric field is effectively a current (called the “displacement current”) there is no paradox.  James Clerk Maxwell (1831-1879) Physics 122B - Lecture 22

23. Electromagnetic Waves Maxwell’s formulation of electricity and magnetism has an interesting consequence. The equations can be manipulated to give a wave equations for E and B of the form: This can be recognized as describing an electromagnetic wave traveling through space with a velocity of: This is quite a remarkable result. Somehow, equations for charges and currents making stationary electric and magnetic fields are telling us about electromagnetic waves traveling through space at the speed of light! Physics 122B - Lecture 22

24. Generators The figure shows a coil with N turns rotating in a magnetic field, with the coil connected to an external circuit by slip rings that transmit current independent of rotation. The flux through the coil is: Therefore, the device produces emf and current that will vary sinusoidally, alternately positive and negative. This is called an alternating current generator, producing what we call AC voltage. Physics 122B - Lecture 22

25. Example: An AC Generator A coil with area 2.0 m2 rotates in a 0.10 T magnetic field at a frequency of 60 Hz. How many turns are needed to generate an AC emf with a peak voltage of 160 V? Physics 122B - Lecture 22

26. Transformers When a coil wound around aniron core is driven by an AC voltageV1cos wt, it produces an oscillatingmagnetic field that will induce anemf V2cos wt in a secondary coilwound on the same core. This iscalled a transformer. The input emf V1 induces acurrent I1 in the primary coil thatis proportional to 1/N1. The flux inthe iron is proportional to this, andit induces an emf V2 in the secondary coil that is proportional to N2. Therefore, V2 = V1(N2/N1). From conservation of energy, assuming no losses in the core, V1I1 = V2I2. Therefore, the currents in the primary and secondary are related by the relation I1 = I2(N2/N1). A transformer with N2>>N1 is called a step-up transformer, which boosts the secondary voltage. A transformer with N2<<N1 is called a step-down transformer, and it drops the secondary voltage. Physics 122B - Lecture 22

27. The Tesla Coil A special case of a step-up transformer is the Tesla coil. It uses no magnetic material, but has a very high N2/N1 ratio and uses high-frequency electrical current to induce very high voltages and very high frequencies in the secondary. There is a phenomenon called “the skin effect” that causes high frequency AC currents to reside mainly on the outer surfaces of conductors. Because of the skin effect, one does not feel (much) the electrical discharges from a Tesla coil. Physics 122B - Lecture 22

28. Metal Detectors Metal detectors like those used at airports can detect any metal objects, not just magnetic materials like iron. They operate by induced currents. A transmitter coil sends high frequency alternating currents that will induce current flow in conductors in its field. Because of Lenz’s Law, the induced current opposes the field from the transmitter, so that net field is reduced. A receiver coil detects the reduction in the magnetic fields from the transmitter and registers the presence of metal. Physics 122B - Lecture 22

29. (Self-) Inductance We define the inductanceL of a coil of wire producing flux Fm as: The unit of inductance is the henry: 1 henry = 1 H = 1 T m2/A = 1 Wb/A The circuit diagram symbol used to represent inductance is: Example: The inductance of a long solenoid with N turns of cross sectional area A and length l is: Physics 122B - Lecture 22

30. Example: Length of an Inductor An inductor is made by tightly winding 0.30 mm diameter wire around a 4.0 mm diameter cylinder. What length cylinder has an inductance of 10 mH? Physics 122B - Lecture 22

31. Potential Across an Inductor Physics 122B - Lecture 22

32. Potential Across an Inductor (2) Physics 122B - Lecture 22

33. Lecture 21 Announcements • Lecture HW has been posted and is due on Wednesday at 10 PM. • Next Friday we will have Exam 3 in this room. It will consist of multiple-choice questions on the Laboratory (25 pts) and Lecture (35 pts). Bring a Scatron sheet, a double-sided page of notes, and a calculator with good batteries. Physics 122B - Lecture 22