Chapter 31: Faraday's Law Notes

Chapter 31: Faraday’s Law

Introduction

• Electricity and magnetism studies have focused on:
  • Electric fields produced by stationary charges.
  • Magnetic fields produced by moving charges.
• This chapter explores effects produced by magnetic fields varying in time.
• Experiments by Michael Faraday (England, 1831) and Joseph Henry (USA, 1831) independently showed that an EMF could be induced in a circuit by a changing magnetic field.
• These experiments led to Faraday’s law of induction, a basic law of electromagnetism.
• An EMF (and therefore a current) can be induced in processes involving changing magnetic flux.

Faraday’s Law of Induction

• Experimental setup: A loop of wire connected to a sensitive ammeter (Figure 1).
• Observations:
  • When a magnet is moved toward the loop, the ammeter reading changes from zero to a nonzero value (negative in Figure 1a).
  • When the magnet is stationary relative to the loop, the ammeter reads zero (Figure 1b).
  • When the magnet is moved away from the loop, the ammeter reading changes to a positive value (Figure 1c).
  • When the magnet is held stationary, and the loop is moved, the ammeter reading changes from zero.
• Conclusion: The loop detects the magnet’s motion relative to it, related to a change in the magnetic field.
• Remarkable result: A current is set up without any batteries in the circuit.
  • This current is called an induced current, produced by an induced EMF.
• Faraday's experiment (Figure 2):
  • A primary coil is wrapped around an iron ring and connected to a switch and a battery. Closing the switch produces a magnetic field.
  • A secondary coil is also wrapped around the ring and connected to an ammeter; it has no battery and is not electrically connected to the primary coil.
  • When the switch in the primary circuit is opened or closed, the ammeter reading in the secondary circuit changes momentarily from zero and then returns to zero. The sign changes depending on whether the switch is opened or closed.
  • The ammeter reads zero when there is either a steady current or no current in the primary circuit.
• Explanation:
  • Closing the switch causes the current in the primary circuit to produce a magnetic field that penetrates the secondary circuit.
  • The changing magnetic field induces a current in the secondary circuit only when the current in the primary circuit changes.
  • No current is induced in the secondary coil when a steady current exists in the primary coil.
• Faraday’s conclusion: An electric current can be induced in a loop by a changing magnetic field.
  • The induced current exists only while the magnetic field through the loop is changing.
  • The loop behaves as though a source of EMF were connected to it for a short time.
  • An induced EMF is produced in the loop by the changing magnetic field.
• Common factor in the experiments: An EMF is induced in a loop when the magnetic flux through the loop changes with time.
  • The EMF is directly proportional to the time rate of change of magnetic flux through the loop.
• Faraday’s law of induction (Equation 1): \varepsilon = -\frac{d\Phi_B}{dt}
  • $\PhiB$ is the magnetic flux through the loop: \PhiB = \int \mathbf{B} \cdot d\mathbf{A}
• For a coil with N loops of the same area, the total induced EMF is (Equation 2): \varepsilon = -N \frac{d\Phi_B}{dt}
• The negative sign has important physical significance (Lenz’s law).
• For a loop enclosing an area A in a uniform magnetic field B (Figure 3), the magnetic flux is BA \cos \theta, where \theta is the angle between the magnetic field and the normal to the loop. Thus, the induced EMF can be expressed as:
  • \varepsilon = - \frac{d}{dt} (BA \cos \theta)
• Induced EMF can be generated by:
  • Changing the magnitude of B with time.
  • Changing the area A enclosed by the loop with time.
  • Changing the angle \theta between B and the normal to the loop with time.
  • Any combination of the above.
• Q1: A circular loop of wire is held in a uniform magnetic field, with the plane of the loop perpendicular to the field lines. Which of the following will not cause a current to be induced in the loop?
  • (c) keeping the orientation of the loop fixed and moving it along the field lines

Applications of Faraday’s Law

• Ground Fault Circuit Interrupter (GFCI) (Figure 4):
  • A safety device protects users against electric shock using Faraday’s law.
  • Wire 1: from wall outlet to appliance.
  • Wire 2: from appliance back to wall outlet.
  • An iron ring surrounds the two wires, and a sensing coil is wrapped around part of the ring.
  • Normally, currents in the wires are in opposite directions and of equal magnitude, so the net current and magnetic flux through the sensing coil are zero.
  • If return current in wire 2 changes (e.g., appliance becomes wet, current leaks to ground), the net current through the ring is no longer zero, and the magnetic flux through the sensing coil is non-zero.
  • Household current is alternating, so the magnetic flux changes with time, inducing an EMF in the coil.
  • This induced EMF triggers a circuit breaker, stopping the current before it reaches a harmful level.
• Electric Guitar Pickup (Figure 5):
  • The coil (pickup coil) is placed near the vibrating guitar string, which is made of a metal that can be magnetized.
  • A permanent magnet inside the coil magnetizes the portion of the string nearest the coil.
  • When the string vibrates, the magnetized segment produces a changing magnetic flux through the coil.
  • The changing flux induces an EMF in the coil that is fed to an amplifier.
  • The amplifier output is sent to loudspeakers, producing the sound waves.
• Q2: A coil consists of 200 turns of wire. Each turn is a square of side d = 18 cm, and a uniform magnetic field directed perpendicular to the plane of the coil is turned on. If the field changes linearly from 0 to 0.50 T in 0.80 s, what is the magnitude of the induced emf in the coil while the field is changing?
  • \varepsilon = N \frac{d\Phi_B}{dt} = N A \frac{dB}{dt} = 200 \times (0.18 \text{ m})^2 \times \frac{0.50 \text{ T}}{0.80 \text{ s}} = 4.05 \text{ V}
• Q3: A loop of wire enclosing an area A is placed in a region where the magnetic field is perpendicular to the plane of the loop. The magnitude of B varies in time according to the expression B = B_0 e^{-at}, where a is some constant. Find the induced emf in the loop as a function of time. (Fig.6)
  • \varepsilon = - \frac{d\PhiB}{dt} = -A \frac{dB}{dt} = -A \frac{d}{dt} (B0 e^{-at}) = A a B_0 e^{-at}

Motional EMF

• Motional EMF refers to the EMF induced in a conductor moving through a constant magnetic field.

• Setup (Figure 7): A straight conductor of length 𝑙 moving through a uniform magnetic field B (directed into the page) with constant velocity v perpendicular to the field.

• The electrons in the conductor experience a magnetic force: \mathbf{F}_B = q \mathbf{v} \times \mathbf{B} (Eq. 29.1)

  • This force is directed along the length 𝑙, perpendicular to both v and B.
  • Electrons move to the lower end of the conductor, leaving a net positive charge at the upper end, creating an electric field E inside the conductor.
  • Charges accumulate until the downward magnetic force qvB is balanced by the upward electric force qE.

• Equilibrium condition: qE = qvB or E = vB

• The potential difference across the ends of the conductor: \Delta V = E \ell = B \ell v (Equation 4)

  • The upper end is at a higher electric potential than the lower end.
  • A potential difference is maintained as long as the conductor moves through the field.
  • Reversing the motion reverses the polarity of the potential difference.

• Moving Conductor in a Closed Conducting Path (Figure 8a):

  • A conducting bar of length 𝑙 slides along two fixed, parallel conducting rails with resistance R.
  • A uniform and constant magnetic field B is applied perpendicular to the plane of the circuit.
  • The bar is pulled to the right with velocity v under an applied force F_{app}.
  • Free charges in the bar experience a magnetic force, setting up an induced current in the closed conducting path.

• The magnetic flux through the circuit is \Phi_B = B \ell x, where x is the position of the bar.

• Induced motional EMF (Equation 5): \varepsilon = - \frac{d\Phi_B}{dt} = -B \ell \frac{dx}{dt} = -B \ell v

• The magnitude of the induced current (Equation 6): I = \frac{\varepsilon}{R} = \frac{B \ell v}{R}

• Energy considerations: The applied force does work on the conducting bar, with the input energy appearing as internal energy in the resistor.

• The magnetic force on the bar is F_B = I \ell B.

• Because the bar moves with constant velocity, the applied force F_{app} is equal in magnitude and opposite in direction to the magnetic force.

• The power delivered by the applied force (Equation 7): P = F_{app} v = I \ell B v = \frac{B^2 \ell^2 v^2}{R}

  • This power input is equal to the rate at which energy is delivered to the resistor, consistent with the principle of conservation of energy.

• Q4: In Figure 31.8a, a given applied force of magnitude F_{app} results in a constant speed v and a power input P. Imagine that the force is increased so that the constant speed of the bar is doubled to 2v. Under these conditions, what are the new force and the new power input?

  • (c) 2F and 4P

• Q5: The conducting bar illustrated in Figure 31.9 moves on two frictionless, parallel rails in the presence of a uniform magnetic field directed into the page. The bar has mass m, and its length is 𝑙. The bar is given an initial velocity v_i to the right and is released at t = 0.

• (A) Using Newton’s laws, find the velocity of the bar as a function of time.

• (B) Show that the same result is found by using an energy approach.

• (C) Suppose you wished to increase the distance through which the bar moves between the time it is initially projected and the time it essentially comes to rest. You can do so by changing one of three variables (v_i, R, or B) by a factor of 2 or 1/2. Which variable should you change to maximize the distance, and would you double it or halve it?

• Q6: A conducting bar of length 𝑙 rotates with a constant angular speed 𝑣 about a pivot at one end. A uniform magnetic field B is directed perpendicular to the plane of rotation as shown in Figure 31.10. Find the motional emf induced between the ends of the bar.

Lenz’s Law

• Faraday’s law indicates that the induced EMF and the change in flux have opposite algebraic signs.

• Lenz’s law: The induced current in a loop is in the direction that creates a magnetic field that opposes the change in magnetic flux through the area enclosed by the loop.

  • The induced current tends to keep the original magnetic flux through the loop from changing.
  • This law is a consequence of the law of conservation of energy.

• Example: A bar moving to the right on two parallel rails in a magnetic field (Figure 11a).

  • As the bar moves to the right, the magnetic flux through the area increases.
  • Lenz’s law states that the induced current must be directed to produce a field directed out of the page (counterclockwise) to oppose this change.
  • If the bar is moving to the left (Figure 11b), the external magnetic flux decreases.
  • The induced current must be clockwise to produce a field directed into the page.
  • In either case, the induced current attempts to maintain the original flux through the area enclosed by the current loop.

• If the current were clockwise, the magnetic force exerted on the bar would be to the right, accelerating the rod, increasing its velocity, and increasing the induced current, leading to a violation of the law of conservation of energy.

• Q7: Figure 31.12 shows a circular loop of wire falling toward a wire carrying a current to the left. What is the direction of the induced current in the loop of wire?

  • (a) Clockwise

• Q8: A magnet is placed near a metal loop as shown in Figure 31.13a

  • (A) Find the direction of the induced current in the loop when the magnet is pushed toward the loop.
  • (B) Find the direction of the induced current in the loop when the magnet is pulled away from the loop.

• Q9: A rectangular metallic loop of dimensions 𝑙 and w and resistance R moves with constant speed v to the right as in Figure 31.14a. The loop passes through a uniform magnetic field B directed into the page and extending a distance 3w along the x axis. Define x as the position of the right side of the loop along the x axis.

  • (A) Plot the magnetic flux through the area enclosed by the loop as a function of x.
  • (B) Plot the induced motional emf in the loop as a function of x.
  • (C) Plot the external applied force necessary to counter the magnetic force and keep v constant as a function of x.

Induced EMF and Electric Fields

• A changing magnetic flux induces an EMF and a current in a conducting loop.

  • An electric field is created in the conductor due to the changing magnetic flux.

• Even in the absence of a conducting loop, a changing magnetic field generates an electric field in empty space.

  • This induced electric field is nonconservative, unlike the electrostatic field produced by stationary charges.

• For a conducting loop of radius r in a uniform magnetic field changing with time, the induced EMF is: \varepsilon= - \frac{d\varphi}{dt}

• The work done by the electric field in moving a charge q once around the loop is q \varepsilon.

  • The electric force acting on the charge is qE, and the work done is qE (2\pi r).

• Therefore, q \varepsilon = qE (2 \pi r), which implies \varepsilon = E (2 \pi r)

• Using \varepsilon = - \frac{d \PhiB}{dt} and \PhiB = BA = B \pi r^2, the induced electric field is (Equation 8): E = \frac{1}{2 \pi r} \frac{d \Phi_B}{dt}

• The EMF for any closed path can be expressed as the line integral of E. d s over that path:∮𝐵. 𝑑𝑠.

• Faraday’s law of induction in general form (Equation 9): \varepsilon = \oint \mathbf{E} \cdot d\mathbf{s} = - \frac{d \Phi_B}{dt}

• The induced electric field E is a nonconservative field generated by a changing magnetic field.

  • The field E cannot be an electrostatic field because the line integral of an electrostatic field over a closed loop is zero.

• Q10: A long solenoid of radius R has n turns of wire per unit length and carries a time-varying current that varies sinusoidally as I = I0 \cos \omega t, where I0 is the maximum current and \omega is the angular frequency of the alternating current source (Fig.31.16).

• (A) Determine the magnitude of the induced electric field outside the solenoid at a distance r ˃ R from its long central axis.

• (B) What is the magnitude of the induced electric field inside the solenoid, a distance r from its axis?

Generators and Motors

• Electric generators take in energy by work and transfer it out by electrical transmission.

• Alternating Current (AC) Generator (Figure 17a):

  • A loop of wire rotated by some external means in a magnetic field.
  • In commercial power plants, energy to rotate the loop comes from various sources (hydroelectric, coal-fired, etc.).
  • As the loop rotates, the magnetic flux changes with time, inducing an EMF and a current.

• For a coil with N turns, the magnetic flux is: \Phi_B = NBA \cos \theta = NBA \cos (\omega t)

• The induced EMF (Equation 10): \varepsilon = -N \frac{d \Phi_B}{dt} = NBA \omega \sin (\omega t)

  • The EMF varies sinusoidally with time (Figure 17b).

• The maximum EMF (Equation 11): \varepsilon_{max} = NBA \omega

• The EMF is maximum when \omega t = 90^\circ or 270^\circ, and zero when \omega t = 0^\circ or 180^\circ.

• Commercial generators in the United States and Canada operate at 60 Hz, while in some European countries, they operate at 50 Hz.

• Q11: In an AC generator, a coil with N turns of wire spins in a magnetic field. Of the following choices, which does not cause an increase in the emf generated in the coil?

  • (a) replacing the coil wire with one of lower resistance

• Q12: The coil in an AC generator consists of 8 turns of wire, each of area A = 0.090 0 m2, and the total resistance of the wire is 12.0 Ω. The coil rotates in a 0.500-T magnetic field at a constant frequency of 60.0 Hz.

• (A) Find the maximum induced emf in the coil.

• (B) What is the maximum induced current in the coil when the output terminals are connected to a low-resistance conductor?

• Direct-Current (DC) Generator (Figure 19a):

  • Similar to AC generator, but uses a split ring called a commutator.
  • The output voltage always has the same polarity and pulsates with time (Figure 19b).
  • The contacts to the split ring reverse their roles every half cycle, keeping the polarity of the output voltage the same.
  • Commercial DC generators use many coils and commutators to obtain a steadier DC current.

• Motor:

  • A device into which energy is transferred by electrical transmission while energy is transferred out by work.
  • Essentially a generator operating in reverse: a current is supplied to the coil, and the torque causes it to rotate.
  • As the coil rotates, the changing magnetic flux induces a back EMF, which acts to reduce the current in the coil.
  • The back EMF increases in magnitude as the rotational speed increases.

• The voltage available to supply current equals the difference between the supply voltage and the back EMF.

  • When a motor is turned on, there is initially no back EMF, and the current is very large.
  • As the coil rotates, the back EMF opposes the applied voltage, and the current decreases.
  • A heavy load slows the motor, decreasing the back EMF and increasing the current from the external voltage source.
  • If a motor is jammed, the lack of a back EMF can lead to dangerously high current.

• Modern Application: Hybrid Drive Systems in Automobiles (Figure 20):

  • Combine a gasoline engine and an electric motor to increase fuel economy and reduce emissions.
  • The electric motor accelerates the vehicle from rest until about 15 mi/h (24 km/h) without using the gasoline engine.
  • At higher speeds, the motor and engine work together.
  • When braking, the motor acts as a generator, returning some of the vehicle’s kinetic energy to the battery.

• Q13: A motor contains a coil with a total resistance of 10 Ω and is supplied by a voltage of 120 V. When the motor is running at its maximum speed, the back emf is 70 V.

• (A) Find the current in the coil at the instant the motor is turned on.

• (B) Find the current in the coil when the motor has reached maximum speed.