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

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. 12/2/2018 Physics 122B - Lecture 22

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; 12/2/2018 Physics 122B - Lecture 22

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 area Aeff = A cos q of the loop We use these ideas to define the magnetic flux: 12/2/2018 Physics 122B - Lecture 22

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: 12/2/2018 Physics 122B - Lecture 22

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? 12/2/2018 Physics 122B - Lecture 22

Magnetic Flux in 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: 12/2/2018 Physics 122B - Lecture 22

Example: Magnetic Flux from a 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? 12/2/2018 Physics 122B - Lecture 22

Heinrich Friedrich Emil Lenz 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. 12/2/2018 Physics 122B - Lecture 22

Lenz’s Law (2) If the field of the bar magnet is already in the loop and the bar magnet is removed, the induced current is in the direction that tries to keep the field constant. If the loop is a superconductor, a persistent standing current is induced in the loop, and the field remains constant. Superconducting loop 12/2/2018 Physics 122B - Lecture 22

Six Induced Current Scenarios 12/2/2018 Physics 122B - Lecture 22

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? 12/2/2018 Physics 122B - Lecture 22

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? 12/2/2018 Physics 122B - Lecture 22

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. 12/2/2018 Physics 122B - Lecture 22

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. 12/2/2018 Physics 122B - Lecture 22

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. 12/2/2018 Physics 122B - Lecture 22

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. 12/2/2018 Physics 122B - Lecture 22

What does Faraday’s Law 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: motional emf new physics The second term says that an emf can be created simply by changing a magnetic field, even if nothing is moving. 12/2/2018 Physics 122B - Lecture 22

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. 12/2/2018 Physics 122B - Lecture 22

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. 12/2/2018 Physics 122B - Lecture 22

Induced Fields and Electromagnetic Waves There is still a puzzle piece missing. Faraday’s Law allows us to calculate an induced current, but what causes the current? What force 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. 12/2/2018 Physics 122B - Lecture 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 through the current (a) or through the capacitor gap (b). If the changing electric field is effectively a current (called the “displacement current”) there is no paradox.  James Clerk Maxwell (1831-1879) 12/2/2018 Physics 122B - Lecture 22

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! 12/2/2018 Physics 122B - Lecture 22

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. 12/2/2018 Physics 122B - Lecture 22

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? 12/2/2018 Physics 122B - Lecture 22

Transformers When a coil wound around an iron core is driven by an AC voltage V1cos wt, it produces an oscillating magnetic field that will induce an emf V2cos wt in a secondary coil wound on the same core. This is called a transformer. The input emf V1 induces a current I1 in the primary coil that is proportional to 1/N1. The flux in the iron is proportional to this, and it 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. 12/2/2018 Physics 122B - Lecture 22

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. 12/2/2018 Physics 122B - Lecture 22

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. 12/2/2018 Physics 122B - Lecture 22

(Self-) Inductance We define the inductance L 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: 12/2/2018 Physics 122B - Lecture 22

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? 12/2/2018 Physics 122B - Lecture 22

Potential Across an Inductor 12/2/2018 Physics 122B - Lecture 22

Potential Across an Inductor (2) 12/2/2018 Physics 122B - Lecture 22

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. 12/2/2018 Physics 122B - Lecture 22