Electromagnetic Induction and Faraday's Law Study Guide

Fundamentals of Induced EMF

Historical Discovery: - Almost 200 years ago, Michael Faraday investigated whether a magnetic field could induce an electric current. - Using a specific apparatus consisting of coils and a switch, Faraday observed that a steady magnetic field (from a steady current) produced no evidence of an induced current. - He discovered that a current was induced only when the magnetic flux was changing, specifically when the switch was being turned on or off.
Mechanism of Induction: - A changing magnetic field induces an electromotive force (EMF). - This change can occur because the current producing the field is changing (as in Faraday's switch experiment) or because a magnet is physically moving relative to a wire loop.

Faraday’s Law of Induction and Lenz’s Law

Magnetic Flux ( $\Phi_B$ ): - Magnetic flux is defined mathematically as the product of the magnetic field and the area through which it passes, adjusted by the angle of orientation. - Equation: $\Phi_B = B A \cos(\theta)$ . - The angle $\theta$ is measured between the magnetic field $\mathbf{B}$ and the area vector $\mathbf{A}$ (which is perpendicular to the face of the loop). - Analogy: Magnetic flux is proportional to the total number of magnetic field lines passing through the loop, similar to electric flux.
Unit of Magnetic Flux: - The SI unit is the weber (Wb). - $1\,\text{Wb} = 1\,\text{T}\cdot\text{m}^2$ .
Faraday’s Law of Induction: - The EMF ( $\mathcal{E}$ ) induced in a circuit is equal to the rate of change of magnetic flux through the circuit. - Equation for a single loop: $\mathcal{E} = -\frac{d\Phi_B}{dt}$ . - Equation for a coil with $N$ loops: $\mathcal{E} = -N \frac{d\Phi_B}{dt}$ .
Lenz’s Law: - The minus sign in Faraday’s law indicates the direction of the induced EMF. - Definition: An induced EMF is always in a direction that opposes the original change in flux that caused it. - Phenomenon: A current produced by an induced EMF moves such that the magnetic field it generates tends to restore/oppose the field changes.
Conditions for Changing Magnetic Flux: - Flux changes if the magnetic field magnitude ( $B$ ) changes. - Flux changes if the area ( $A$ ) of the loop changes. - Flux changes if the angle ( $\theta$ ) between the loop and the field changes.

Problem Solving with Lenz’s Law

Determine Flux Change: Identify if the magnetic flux is increasing, decreasing, or unchanged.
Determine Induced Field Direction: - If flux is increasing: The magnetic field due to the induced current points in the opposite direction to the original field. - If flux is decreasing: The induced field points in the same direction as the original field to maintain the flux. - If flux is unchanged: The induced field is zero.
Determine Current Direction: Use the right-hand rule to find the current direction based on the induced field.
Field Distinction: Remember that the external magnetic field and the field created by the induced current are distinct entities.

EMF Induced in a Moving Conductor (Motional EMF)

Motional EMF Principles: - Moving a conducting bar through a magnetic field causes a change in the area of the circuit, thereby changing flux. - The induced current creates a force that tends to slow the moving bar. - To maintain constant motion, an external force must be applied to the bar.
Mathematical Model: - Magnitude of induced EMF: $\mathcal{E} = B l v$ . - This formula is valid when the magnetic field ( $B$ ), length of the conductor ( $l$ ), and velocity ( $v$ ) are mutually perpendicular. If not, only their perpendicular components are used.

Mathematical Examples in Induction

Example 29-2: A loop of wire in a magnetic field: - A square loop has side $l = 5.0\,\text{cm}$ ( $A = 0.0025\,\text{m}^2$ ) in a uniform field $B = 0.16\,\text{T}$ . - (a) Flux when $B$ is perpendicular to the face ( $\theta = 0^{\circ}$ ): $\Phi_B = B A = (0.16\,\text{T})(0.0025\,\text{m}^2) = 4.0 \times 10^{-4}\,\text{Wb}$ . - (b) Flux when $B$ is at an angle of $30^{\circ}$ to the area vector $\mathbf{A}$ : $\Phi_B = B A \cos(30^{\circ})$ . - (c) Average current ( $I$ ) if resistance $R = 0.012\,\Omega$ and the loop rotates from (b) to (a) in $0.14\,\text{s}$ .
Example 29-5: Pulling a coil from a magnetic field: - Square coil: $100\text{ loops}$ , side $l = 5.00\,\text{cm}$ , resistance $R = 100\,\Omega$ , field $B = 0.600\,\text{T}$ . - Coil is pulled out of the field in $0.100\,\text{s}$ . - Calculations: Rate of change in flux, induced EMF, induced current, energy dissipated, and average external force required ( $F_{ext}$ ).
Example 29-6: Moving Airplane EMF: - Speed $v = 1000\,\text{km/h}$ ( $\approx 278\,\text{m/s}$ ), Earth's vertical field $B = 5 \times 10^{-5}\,\text{T}$ , wingspan $L = 70\,\text{m}$ . - Potential difference induced between wing tips: $\mathcal{E} = B L v$ .
Example 29-7: Electromagnetic Blood-Flow Measurement: - Used to measure flow velocity in vessels because blood contains charged ions. - Parameters: vessel diameter $d = 2.0\,\text{mm}$ , $B = 0.080\,\text{T}$ , measured EMF $\mathcal{E} = 0.10\,\text{mV}$ . - Blood velocity $v$ can be solved using $\mathcal{E} = B d v$ .

Electric Generators

Function: A generator is the functional opposite of a motor; it transforms mechanical energy into electrical energy.
AC Generator Components: - An axle rotated by external forces (falling water, steam). - Slip rings and brushes that maintain electrical contact while the loop rotates.
EMF Production: - If a loop rotates with constant angular velocity $\omega$ , the flux changes sinusoidally. - For a coil with $N$ loops, the induced EMF is: $\mathcal{E} = N B A \omega \sin(\omega t)$ .

Transformers and Power Transmission

Transformer Construction: Consists of a primary and a secondary coil, either interwoven or linked by an iron core.
Voltage Transformation: - A changing EMF in the primary coil induces an EMF in the secondary coil via magnetic flux linkage. - Transformer Equation: $\frac{V_s}{V_p} = \frac{N_s}{N_p}$ . - Step-up Transformer: Secondary voltage is higher than primary (N_s > N_p). - Step-down Transformer: Secondary voltage is lower than primary (N_s < N_p).
Energy Conservation and Efficiency: - In an ideal transformer (no losses), power is conserved ( $P_p = P_s$ ). - Current Ratio: $\frac{I_s}{I_p} = \frac{N_p}{N_s}$ . The current ratio is the inverse of the turns/voltage ratio.
Example 29-12: Cell Phone Charger: - Reduces $120\,\text{V}$ AC to $5.0\,\text{V}$ AC (then rectified to $5.0\,\text{V}$ DC for a $3.7\,\text{V}$ battery). - Given: secondary loops $N_s = 30$ , secondary current $I_s = 700\,\text{mA}$ . - Required: Determine primary turns ( $N_p$ ), primary current ( $I_p$ ), and power transformed ( $P$ ).
Transmission of Power: - Transformers require alternating current (AC) because they rely on changing flux. - Power is transmitted over long distances at high voltages to minimize power loss ( $P_{loss} = I^2 R$ ). - Example 29-13: Transmission Lines: - Power $P = 120\,\text{kW}$ , distance $10\,\text{km}$ , total resistance $R = 0.40\,\Omega$ . - Loss at $240\,\text{V}$ : High current leads to significant loss. - Loss at $24,000\,\text{V}$ : Low current dramatically reduces heat/power loss.

Practical Applications of Induction

Induction Stove: - An AC current in a coil (the burner) produces a changing magnetic field. - It heats metal pans (conductors) due to induced currents but will not heat glass (insulator).
Sound Systems (Microphone): A vibrating membrane attached to a coil moves relative to a magnet, inducing an EMF that corresponds to sound waves.
Computer Memory: Magnetic storage relies on induction principles for reading/writing data.
Seismograph: Measures Earth tremors using a fixed coil and a magnet hung on a spring (or vice versa); Earth movement induces a current recorded for data.
Ground Fault Circuit Interrupter (GFCI): - Detects current imbalances between hot and neutral wires using induction. - Interrupts the circuit extremely quickly to prevent electrocution, acting faster than standard circuit breakers.