# Induced Voltages and Inductance

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Induced Voltages and Inductance
Chapter 20 Induced Voltages and Inductance Conceptual questions: 1,2,4,6,12,13 Quick Quizzes: 1,3,5 Problems: 26, 28, 34, 39,56

Induced emf A current can be produced by a changing magnetic flux

Magnetic Flux A loop of wire is in a uniform magnetic field B
The loop has an area A The flux is defined as ΦB = BA = B A cos θ θ is the angle between B and the normal to the plane SI units of flux are T m² = Wb (Weber)

Magnetic Flux The value of the magnetic flux is proportional to the total number of lines passing through the loop When the area is perpendicular to the lines, the maximum number of lines pass through the area and the flux is a maximum When the area is parallel to the lines, no lines pass through the area and the flux is 0

Faraday’s Law For a single loop For N tightly wound up loops Since ΦB = B A cos θ, the change in the flux, ΔΦ, can be produced by a change in B, A or θ. Thus, the induced electromotive force can be produced by changing B, A or θ, or their combinations.

emf induced by changing magnetic field

QUICK QUIZ 20.1 The figure below is a graph of magnitude B versus time t for a magnetic field that passes through a fixed loop and is oriented perpendicular to the plane of the loop. Rank the magnitudes of the emf generated in the loop at the three instants indicated (a, b, c), from largest to smallest.

Motional emf, changing A
A straight conductor of length ℓ moves perpendicularly with constant velocity through a uniform field The electrons in the conductor experience a magnetic force F = q v B The electrons tend to move to the lower end of the conductor

Motional emf, cont The potential difference between the ends of the conductor can be found by ΔV = B ℓ v A potential difference is maintained across the conductor as long as there is motion through the field If the motion is reversed, the polarity of the potential difference is also reversed

Motional emf in a Circuit
The induced, motional emf, acts like a battery in the circuit

QUICK QUIZ 20.3 You wish to move a rectangular loop of wire into a region of uniform magnetic field at a given speed so as to induce an emf in the loop. The plane of the loop must remain perpendicular to the magnetic field lines. In which orientation should you hold the loop while you move it into the region of magnetic field in order to generate the largest emf? (a) With the long dimension of the loop parallel to the velocity vector; (b) With the short dimension of the loop parallel to the velocity vector. (c) Either way—the emf is the same regardless of orientation.

The negative sign in Faraday’s Law is included to indicate the polarity of the induced emf, which is found by Lenz’ Law The polarity of the induced emf is such that it produces a current whose magnetic field opposes the change in magnetic flux through the loop The induced current tends to maintain the original flux through the circuit

Lenz’ Law Revisited – Moving Bar Example
As the bar moves to the right, the magnetic flux through the circuit increases with time because the area of the loop increases The induced current must in a direction such that it opposes the change in the external magnetic flux

Lenz’ Law, Bar Example The bar is moving toward the left
The magnetic flux through the loop is decreasing with time The induced current must be clockwise to to produce its own flux into the page

Lenz’ Law, Moving Magnet Example
A bar magnet is moved to the right toward a stationary loop of wire (a) As the magnet moves, the magnetic flux increases with time The induced current produces a flux to the left, so the current is in the direction shown (b)

Lenz’ Law, Final Note When applying Lenz’ Law, there are two magnetic fields to consider The external changing magnetic field that induces the current in the loop The magnetic field produced by the current in the loop

Conceptual questions A circular loop is located in a uniform and constant magnetic field. Describe how an emf can be induced in the loop. Does dropping a magnet down a copper tube produce a current in the tube? 12. A bar magnet is dropped toward a conducting ring lying on a floor. As the magnet falls toward the ring, does it move as a freely falling body? 4. A loop of wire is placed in a uniform magnetic field. For what orientation of the loop is the magnetic flux a maximum? For what orientation is it zero? 6. A bar moves perpendicularly to the magnetic field. Is an external force required to keep it moving with a constant velocity?

What is the direction of the current induced in the resistor at the instant the switch is closed?
Problem I B I B Induced current Induced B

20-28. Find the direction of the current in R the instant the switch is closed.

Applications of Faraday’s Law – Electric Guitar
A vibrating string induces an emf in a coil A permanent magnet inside the coil magnetizes a portion of the string nearest the coil As the string vibrates at some frequency, its magnetized segment produces a changing flux through the pickup coil The changing flux produces an induced emf that is fed to an amplifier

Applications of Faraday’s Law – Ground Fault Interrupters

Tape Recorder A magnetic tape moves past a recording and playback head
The tape is a plastic ribbon coated with iron oxide or chromium oxide To record, the sound is converted to an electrical signal which passes to an electromagnet that magnetizes the tape in a particular pattern To playback, the magnetized pattern is converted back into an induced current driving a speaker

Recording

Tape playing

AC Generators As the loop rotates, the magnetic flux through it changes with time This induces an emf and a current in the external circuit The ends of the loop are connected to slip rings that rotate with the loop Connections to the external circuit are made by stationary brushed in contact with the slip rings

AC Generators If the loop rotates with a constant angular speed, ω, and N turns ε = N B A ω sin ω t ε = εmax = NBAω when loop is parallel to the field ε = 0 when when the loop is perpendicular to the field

Problem A coil of area 0.10 m2 is rotating at 60 rev/s with its axis of rotation perpendicular to a 0.20-T magnetic field. (a) If there are turns on the loop, what is the maximum voltage induced in the coil? (b) When the maximum induced voltage occurs, what is the orientation of the loop with respect to the magnetic field? Using, emax = NBA w =1000 (0.2T) (0.10m2) (120 p rad/s) = V Plane of the loop is parallel to the magnetic field

DC Generators Components are essentially the same as that of an ac generator The major difference is the contacts to the rotating loop are made by a split ring, or commutator

DC Generators The output voltage always has the same polarity
The current is a pulsing current

Self-inductance Self-inductance occurs when the changing flux through a circuit arises from the circuit itself As the current increases, the magnetic flux through a loop due to this current also increases The increasing flux induces an emf that opposes the current As the magnitude of the current increases, the rate of increase lessens and the induced emf decreases This opposing emf results in a gradual increase of the current

Self-inductance cont The self-induced emf must be proportional to the time rate of change of the current L is inductance of the device, unit Henry 1 H = 1 (V · s) / A

Self inductance of a solenoid
= BA cos q, when q =90o F = BA [A is the cross-sectional area of the solenoid] For a solenoid B = mo n I = mo (N/l)I, F = mo A NI/l L = N F /I = mo A N2 /l L depends only on geometric factors A and l, number of turns squared, and on mo

Problem 20.39 A solenoid of radius 2.5 cm has 400 turns and a length of 20 cm. Find (a) its inductance and (b) the rate at which current must change through it to produce an emf of 75 mV.

QUICK QUIZ 20.5 The switch in the circuit shown in the figure below is closed and the lightbulb glows steadily. The inductor is a simple air-core solenoid. An iron rod is inserted into the interior of the solenoid, which increases the magnitude of the magnetic field in the solenoid. As the rod is inserted into the solenoid, the brightness of the lightbulb (a) increases, (b) decreases, or (c) remains the same.

Energy Stored in a Magnetic Field
PEL = ½ L I2

Conceptual question 13. If the current in an inductor is doubled, by what factor does the stored energy change?

Problem 20-56 A novel method of storing electrical energy has been proposed. A huge underground superconducting coil, 1.00 km in diameter, would be fabricated. It would carry a maximum current of 50.0 kA through each winding of a 150-turn Nb3Sn solenoid. If the inductance of this huge coil is 50.0 H, what is the total energy stored? (b) What is the compressive force per meter length acting between two adjacent windings m apart? (Hint: Because the radius of the coil is so large, the magnetic field created by one winding and acting on an adjacent turn can be considered to be that of a long, straight wire.)

Review questions 1. A heavy permanent magnet is moving toward a current carrying circular loop of wire. Which is correct? The coil will push or pull the magnet just as hard as the magnet pulls or pushes the coil. The magnet pushes harder on the coil than the coil pushes on the magnet because the magnet is more massive than the coil. The magnet will push or pull on the coil, but the coil will not push or pull on the magnet at all because the coil is not a magnet.

A conducting bar is sliding at a constant velocity along two conducting horizontal rods. The rods are separated by a distance l and connected across by a resistor R. The entire apparatus is placed inside a magnetic field B directed into the page. How will the current in the apparatus be generated? a. sinusoidally b. clockwise c. counterclockwise d. not enough information v

3. A conducting coil is rotated at a constant speed in an external magnetic field. Which of the following most likely represents the current generated within the coil as a function of time? t i t i c d a t i t i b