Chapter 23: Magnetic Flux and Faraday's Law of Induction

Introduction to Electromagnetic Induction

Fundamental Question: While it is established that electric current produces a magnetic field, the inverse possibility is explored: Can a magnetic field produce an electric current?
Definition: The phenomenon where a magnetic field generates an electric current is defined as Electromagnetic Induction.

Faraday’s Experimental Observations

Experiment 1: The Bar Magnet and Coil: * When a bar magnet is pushed into a stationary coil of wire, a momentary deflection is observed on the needle of a current-meter. * If the magnet is held stationary inside the coil, no effect (no current) is observed. * A quick withdrawal of the magnet from the coil cause the needle to deflect in the opposite direction, indicating a reversal of current flow.
Experiment 2: Moving the Coil: * A momentary current is produced by rapidly pulling a coil of wire out of a magnetic field. * Conversely, pushing the coil into the magnet causes the meter needle to deflect in the opposite direction.
Experiment 3: Two-Coil Interaction: * In a setup where one coil is placed directly above another, constant current in the first coil produces no current in the lower circuit while the switch remains in a closed position. * A momentary current appears in the lower circuit only during the specific instances when the switch in the primary circuit is either opened or closed.

The Contribution of Michael Faraday

Biographical Context: Michael Faraday (1791–1867) was a renowned experimental scientist.
Key Inventions: He is credited with the invention of the electric motor, the generator, and transformers.
The Discovery of 1831: Faraday discovered that an electrical current is produced by a changing magnetic field.

Magnetic Flux

Conceptual Definition: Magnetic flux is a measure of the total magnetic field passing through a given area. It is proportional to the total number of magnetic field lines passing through a loop of wire.
Mathematical Definition: Given a loop of wire with area $A$ in a uniform magnetic field $B$ , the magnetic flux $ext{\Phi}<em>B$ is defined as: * $ext{\Phi}_B = B</em>{\perp}A = BA ext{\cos}( heta)$ * $heta$ is defined as the angle between the magnetic field vector $B$ and the normal (perpendicular vector) to the plane of the loop.
Angular Dependencies: * Maximum Flux: When the field is perpendicular to the plane of the loop, the angle $heta = 0$ . In this state, flux is at its maximum value: $ext{\Phi}<em>B = ext{\Phi}</em>{B, ext{max}} = BA$ . * Zero Flux: When the field is parallel to the plane of the loop, the angle $heta = 90^ ext{\circ}$ , resulting in $ext{\Phi}_B = 0$ . * Negative Flux: Flux can take negative values if the orientation is such that, for example, $heta = 180^ ext{\circ}$ .
SI Units: The unit for magnetic flux is the Tesla-square meter ( $T \cdot m^2$ ), which is defined as the Weber (Wb).
Significance: Electromotive force (emf) is induced by a change in the quantity of magnetic flux rather than simply a change in the magnitude of the magnetic field.

Faraday’s Law of Induction

The Law: The instantaneous emf ( $ext{\epsilon}$ ) induced in a circuit is equal to the negative of the time rate of change of magnetic flux through the circuit.
Formula: * ext{\epsilon} = -N rac{\text{\Delta} \text{\Phi}_B}{\text{\Delta} t} * $N$ represents the number of turns in the coil.
Mechanisms for Inducing Emf: Because flux depends on $B$ , $A$ , and $heta$ , an emf can be induced in three primary ways: 1. Changing the magnitude of the magnetic field ( $B$ ). 2. Changing the size of the area ( $A$ ) enclosed by the loop. 3. Rotating the loop to change the angle ( $\theta$ ) between the field and the loop normal.

Lenz’s Law

Purpose: Lenz's Law is used to determine the polarity (direction) of the induced emf and current, represented by the negative sign in Faraday’s Law.
Definition: The current caused by the induced emf travels in a direction such that it creates a magnetic field with a flux that opposes the change in the original flux through the circuit.
Two Fields to Consider: 1. The external changing magnetic field that induces the current. 2. The magnetic field produced by the induced current itself (denoted as $B_{ ext{ind}}$ ).
Application Rules: * If the external magnetic field is increasing, $B_{ ext{ind}}$ is directed opposite to the external field to oppose the increase. * If the external magnetic field is decreasing, $B_{ ext{ind}}$ is directed in the same direction as the external field to oppose the decrease. * The direction of the induced current is determined using the curled-straight right-hand rule based on the direction of the induced field.
Practical Demonstration: A magnet falling through a copper pipe demonstrates these opposing forces (Eddy currents), slowing the magnet's descent.

Applications of Faraday’s Law

The Electric Guitar: * A permanent magnet inside a "pickup coil" magnetizes the portion of the metal guitar string nearest to it. * As the string vibrates at a specific frequency, its magnetized segment creates a changing magnetic flux through the pickup coil. * This changing flux induces an emf in the coil at the same frequency as the string's vibration. * The induced emf is then fed into an amplifier.
Motional Emf: * A straight conductor of length $\ell$ moves perpendicularly with a constant velocity $v$ through a uniform magnetic field $B$ . * Electrons in the conductor experience a Lorentz force ( $F = qvB$ ), causing them to migrate toward the lower end of the conductor. * This charge separation creates a net positive charge at the upper end and a net negative charge at the base, producing an internal electric field. * Charges accumulate until the downward magnetic force is balanced by the upward electric force. * The resulting potential difference (Motional Emf) is maintained as long as the conductor is in motion: * $ext{\Delta} V = B \ell v$ * If the direction of motion is reversed, the polarity of the potential difference is also reversed.

AC Generators

Basic Operation: An Alternating Current (AC) generator converts mechanical energy into electrical energy using a rotating wire loop in a magnetic field.
Components: * Rotating wire loops. * Slip Rings: Connected to the ends of the loop to rotate with it. * Stationary Brushes: Maintain contact with the slip rings to connect the rotating loop to an external circuit.
Energy Sources: Mechanical rotation can be supplied by falling water (hydroelectric) or steam produced by burning coal.
Mathematical Model of the Rotating Loop: * Emf is generated specifically in segments BC and AD of the loop. * Emf for a single wire segment: $ext{\epsilon} = B \ell v_{\perp} = B \ell v ext{\sin}( heta)$ . * Total emf for the loop: $ext{\epsilon} = 2 B \ell v_{\perp} = 2 B \ell v ext{\sin}( heta)$ . * For a loop rotating with constant angular speed ( $ext{\omega}$ ) and having $N$ turns: * $ext{\epsilon} = NBA\omega ext{\sin}( ext{\omega} t)$ * Operating States: * The emf is at its maximum (\text{\epsilon}_{ ext{max}}) when the loop is parallel to the field. * The emf is zero when the loop is perpendicular to the field.
Example Parameters: A generator may operate at a field magnitude of $10.0\,T$ at a speed of $310\, ext{rpm}$ .

Transformers

Construction: Consists of two coils (Primary and Secondary) wound around a core of soft iron. * Primary Coil: Connected to the input AC voltage source ( $ext{\Delta} V_1$ or $V_p$ ) with $N_1$ (or $N_p$ ) turns. * Secondary Coil: Connected to a resistor/load ( $ext{\Delta} V_2$ or $V_s$ ) with $N_2$ (or $N_s$ ) turns.
The Iron Core: Acts to increase magnetic flux and provides a medium for the flux to pass efficiently from the primary to the secondary coil.
Voltage Relation: Because the rate of change of flux is identical for both coils, the voltages are related by the ratio of turns: * $rac{ ext{\Delta} V_2}{ ext{\Delta} V_1} = rac{N_2}{N_1}$
Types of Transformers: * Step-up Transformer: N_2 > N_1 (Secondary voltage is higher than primary). * Step-down Transformer: N_2 < N_1 (Secondary voltage is lower than primary).

Electrical Power Transmission

Power Conservation: In an ideal transformer, the power input to the primary equals the power output at the secondary: * $I_1 ext{\Delta} V_1 = I_2 ext{\Delta} V_2$
Efficiency: Real transformers have power efficiencies between $90\%$ and $99\%$ .
Transmission Strategy: For long-distance transport, it is most economical to use high voltage and low current. * This minimizes energy losses due to resistance ( $I^2 R$ losses).
Practical Progression: 1. Generating station steps voltage up to approximately $230,000\,V$ . 2. Distribution stations step it down to $20,000\,V$ . 3. Utility poles at customer sites step it down to $120\,V$ .

Self-Inductance

Definition: Self-inductance occurs when the changing magnetic flux through a circuit arises from the circuit itself (due to changes in its own current).
Back Emf: The self-induced emf ( $ext{\epsilon}$ ) opposes the change in current: * $ext{\epsilon} = -L rac{ ext{\Delta} I}{ ext{\Delta} t}$
Inductance ( $L$ ): A proportionality constant that depends on geometric factors of the coil. * Joseph Henry: The unit is named after him; he was also the first director of the Smithsonian. * Formula for L: $L = N rac{ ext{\Delta} ext{\Phi}_B}{ ext{\Delta} I}$ . * Unit: The Henry ( $H$ ). $1\,H = 1\,(V \cdot s) / A$ .

Inductors and RL Circuits

Inductor Function: An inductor serves as a measure of opposition to the rate of change of current (similar to how resistance $R$ opposes the current itself).
Current Lag: As a circuit is closed, the current does not reach its maximum value instantaneously because the inductor produces an emf that opposes the increasing current.
The RL Circuit: * When the current reaches its maximum, the rate of change ( $rac{ ext{\Delta} I}{ ext{\Delta} t}$ ) becomes zero, and thus the back emf becomes zero. * Time Constant (\text{\tau}): The time required for the current in the circuit to reach $63.2\%$ of its final maximum value. * $ext{\tau} = rac{L}{R}$ * Instantaneous Current Formula: * $I = rac{ ext{\epsilon}}{R} (1 - e^{-t/ ext{\tau}})$ (where $t$ is time).