Induced Voltages and Inductance (Chapter 20)

20.1 Induced emf and Magnetic Flux

• Faraday’s Experiment Components:
    - Primary Circuit: Consists of a primary coil connected directly to a battery.
    - Secondary Circuit: Consists of a secondary coil connected to an ammeter. There is no battery in the secondary circuit.

• Experimental Results:
    - When the switch in the primary circuit is closed, the ammeter in the secondary circuit reads a current and then returns to zero.
    - When the switch in the primary circuit is opened again, the ammeter reads a current in the opposite direction and then returns to zero.
    - While there is a steady current in the primary circuit, the ammeter reads zero.

• Conclusions from Faraday’s Experiment:
    - An electrical current is produced by a changing magnetic field.
    - A steady magnetic field does not produce a current unless the coil is moving.
    - Current in the secondary circuit is instantaneous, occurring only while the magnetic field through the secondary coil is changing.
    - The secondary circuit acts as if a source of emf were connected to it for a short time.
    - This phenomenon is described as an induced emf produced by a changing magnetic field.

• Definition of Magnetic Flux:
    - The emf is induced by a change in magnetic flux rather than simply a change in the magnetic field.
    - Magnetic flux ($\Phi_B$) is a measure of the number of magnetic field lines crossing a given area.
    - It is proportional to the strength of the magnetic field passing through the plane of a loop and the area of the loop.
    - The number of field lines per unit area increases as field strength increases.

• Mathematical Definition of Magnetic Flux:
    - For a loop of wire with area $A$ in a uniform magnetic field $B$:
    - $\Phi_B = B_{\perp} A = BA \cos(\theta)$
    - $\theta$ is defined as the angle between the magnetic field and the normal (perpendicular) to the plane of the loop.

• Units:
    - SI unit: weber ($Wb$).
    - Conversion: $1\,Wb = 1\,T\,m^2$.

• Orientation and Flux Magnitude:
    - Maximum Flux: Occurs when field lines are perpendicular to the plane of the loop ($\theta = 0^{\circ}$, because $\cos(0^{\circ}) = 1$).
      - $\Phi_{B,max} = BA$
    - Zero Flux: Occurs when field lines are parallel to the plane of the loop ($\theta = 90^{\circ}$, because $\cos(90^{\circ}) = 0$).
      - $\Phi_B = 0$
    - Negative Flux: Magnetic flux can be negative if $\theta = 180^{\circ}$.

Example 20-1: Magnetic Flux Calculations

• Scenario: A conducting circular loop with radius $r = 0.250\,m$ is in the xy-plane. A uniform magnetic field $B = 0.360\,T$ points in the positive z-direction (same as the normal to the plane).

• (a) Calculate the magnetic flux through the loop:
    - Area $A = \pi r^2 = \pi (0.250\,m)^2$
    - $\Phi_B = BA \cos(0^{\circ}) = (0.360\,T)(\pi)(0.250\,m)^2(1)$

• (b) Loop rotated clockwise around x-axis, normal points at $45.0^{\circ}$ to z-axis:
    - $\theta = 45.0^{\circ}$
    - $\Phi_B = BA \cos(45.0^{\circ}) = (0.360\,T)(\pi)(0.250\,m)^2 \cos(45.0^{\circ})$

• (c) Change in flux ($\Delta \Phi_B$):
    - $\Delta \Phi_B = \Phi_{B,final} - \Phi_{B,initial}$

20.2 Faraday’s Law of Induction and Lenz’s Law

• Electromagnetic Induction Experiment:
    - Moving a magnet toward a loop: Ammeter reads current in one direction.
    - Holding magnet stationary: Ammeter reads zero current.
    - Moving magnet away from loop: Ammeter reads current in opposite direction.
    - Moving the loop toward or away from a stationary magnet produces the same result.
    - Conclusion: Current is set up (induced current) as long as there is relative motion between the magnet and loop.

• Faraday’s Law of Induction:
    - An emf is induced in a circuit when the magnetic flux through the circuit changes with time.
    - The instantaneous emf induced equals the negative rate of change of magnetic flux with respect to time.
    - Formula for a circuit with $N$ loops:
      - $\epsilon = -N \frac{\Delta \Phi_B}{\Delta t}$
    - SI unit: volt ($V$).
    - Change in flux ($\Delta \Phi_B$) can be produced by varying $B$, $A$, or $\theta$.

• Lenz’s Law:
    - Determines the polarity/direction of the induced current.
    - The induced current travels in a direction such that it creates a magnetic field whose flux opposes the change in the original flux through the circuit.
    - Two magnetic fields are involved:
      - 1. The external changing magnetic field.
      - 2. The magnetic field produced by the induced current ($B_{ind}$).
    - Applying Lenz's Law:
      - If external flux is increasing: $B_{ind}$ points in the opposite direction to the external field.
      - If external flux is decreasing: $B_{ind}$ points in the same direction as the external field.
      - If flux is unchanged: Induced current and $B_{ind}$ are zero.

• Right-Hand Rule for Current: Once the direction of $B_{ind}$ is known, use the right-hand rule to find the direction of the induced current.

Examples of Lenz's Law Application

• Magnet toward coil: Induces magnetic field that repels the magnet.

• Magnet away from coil: Induces magnetic field that attracts the magnet.

• Scenario: decreasing outward flux: Induced current will be counterclockwise to create outward field lines.

• Scenario: decreasing inward flux: Induced current will be clockwise to create inward field lines.

• Scenario: increasing inward flux: Induced current will be counterclockwise to oppose the increase.

• Scenario: increasing flux to the left: Induced current will be counterclockwise.

Example 20-2: Induced Current in a Square Coil

• Setup: Coil with $N = 25$ turns, square cross-section $1.80\,cm$ on a side ($A = (0.0180\,m)^2$). Total resistance $R = 0.350\,\Omega$. Magnetic field is perpendicular to the coil.

• (a) Find induced emf if field changes from $0.00\,T$ to $0.500\,T$ in $0.800\,s$:
    - $\Delta \Phi_B = A \times (B_f - B_i) = (0.0180\,m)^2 \times (0.500\,T - 0.00\,T)$
    - $\epsilon = -N \frac{\Delta \Phi_B}{\Delta t}$

• (b) Find magnitude of induced current:
    - $I = \frac{|\epsilon|}{R}$

• (c) Find direction of induced current:
    - Since $B$ is increasing, the induced field must oppose it.

Applications of Faraday’s Law

• Ground Fault Interrupters (GFI): Safety devices protecting against electrical shock by detecting changes in magnetic flux caused by current imbalances.

• Electric Guitar Pickups: A vibrating string induces a voltage in a pickup coil.

• Apnea Monitors: Used for infants to alert caregivers if breathing (which effects flux in a coil) stops.

20.3 Motional emf

• Definition: The potential difference produced in a straight conductor of length $l$ moving with velocity $v$ through a uniform magnetic field $B$ (where $v \perp B$).

• Mechanism:
    - Electrons in the conductor experience a magnetic force $F_m = qvB$ directed toward one end (e.g., downward).
    - Charge separation occurs: electrons accumulate at one end (negative), while the other becomes positive.
    - This separation creates an internal electric field ($E$).
    - Equilibrium is reached when electric force $qE$ balances magnetic force $qvB$.

• Mathematical Relationships:
    - The potential difference is $\Delta V = El = Blv$.
    - If the moving bar is part of a closed circuit with resistance $R$:
      - $\epsilon = Blv$
      - Induced current $I = \frac{\epsilon}{R} = \frac{Blv}{R}$

• Conservation of Energy:
    - Moving the bar through the field requires an applied force to overcome the magnetic force on the induced current.
    - Power delivered to the resistor: $P = I^2 R$.
    - Work done by the applied force equals the energy dissipated in the resistor.

Example 20-4: Sliding Bar Motional emf

• Data: $l = 0.500\,m$, $v = 2.00\,m/s$, $B = 0.250\,T$, $R = 0.500\,\Omega$.

• (a) Induced voltage: $\epsilon = Blv = (0.250\,T)(0.500\,m)(2.00\,m/s) = 0.250\,V$.

• (b) Current and Power:
    - $I = \frac{\epsilon}{R} = \frac{0.250\,V}{0.500\,\Omega} = 0.500\,A$.
    - $P = I^2 R = (0.500\,A)^2(0.500\,\Omega) = 0.125\,W$.

• (c) Magnetic force on bar:
    - $F_m = BIl = (0.250\,T)(0.500\,A)(0.500\,m) = 0.0625\,N$.

• (d) Applied force via power:
    - $P = F_{app} v \implies F_{app} = \frac{P}{v} = \frac{0.125\,W}{2.00\,m/s} = 6.25 \times 10^{-2}\,N$.

20.4 Generators and Motors

• AC (Alternating Current) Generators:
    - Convert mechanical energy to electrical energy.
    - Construction: A wire loop rotated in a magnetic field. Heat sources (coal) or falling water (hydroelectric) turn turbines to rotate the loop.
    - Components: Slip rings (rotate with the loop) and stationary brushes (connect to external circuit).
    - Operation: As the loop rotates, flux changes.
      - $\epsilon = \epsilon_{max}$ when the loop plane is parallel to $B$.
      - $\epsilon = 0$ when the loop plane is perpendicular to $B$.
    - Formula: $\epsilon = NBA \omega \sin(\omega t)$, where $\omega$ is constant angular speed.

• DC (Direct Current) Generators:
    - Essentially the same components as AC generators.
    - Major Difference: Uses a split ring or commutator instead of slip rings.
    - Output: Pulsating DC current (always same polarity). To produce steady DC, multiple loops and commutators are used to superimpose multiple outputs.

• Motors:
    - Convert electrical energy into mechanical energy (generator run in reverse).
    - Back emf: An induced emf that tends to reduce the applied current as the motor coil rotates.
    - Initially, when a motor is turned on, there is no back emf, leading to a very large starting current.
    - As rotation speed increases, back emf opposes the applied voltage, reducing the current.

20.5 Self-Inductance

• Definition: Self-induction occurs when the changing flux through a circuit arises from the circuit itself (due to its own changing current).

• Mechanism: As current increases, the magnetic flux it produces also increases. This induces an emf that opposes the change, resulting in a gradual rather than instantaneous current increase.

• Formula for Self-Induced emf:
    - $\epsilon = -L \frac{\Delta I}{\Delta t}$
    - $L$ is the proportionality constant called self-inductance.
    - SI unit: henry ($H$). $1\,H = 1\,V\,s/A$.

• Inductance Formula for a Solenoid:
    - $L = \frac{N \Delta \Phi_B}{\Delta I}$
    - $L = \frac{\mu_0 N^2 A}{l}$, where $N$ is turns, $A$ is area, and $l$ is length.

• Example 20-7: Solenoid Inductance:
    - Given: $N = 300$, $l = 25.0\,cm$, $A = 4.00 \times 10^{-4}\,m^2$.
    - $\epsilon$ calculation if $I$ decreases at $50.0\,A/s$ ($\frac{\Delta I}{\Delta t} = -50.0\,A/s$).

20.6 RL Circuits

• Inductors: Circuit elements with high inductance, represented by a coil symbol.

• Inductanc e as Opposition: While resistance $R$ opposes current magnitude, inductance $L$ opposes the rate of change of current.

• Current Behavior: Current does not reach the maximum value $\epsilon / R$ instantaneously.

• Time Constant ((\tau)):
    - $\tau = \frac{L}{R}$
    - The time required for the current to reach $63.2\%$ of its final value ($\epsilon / R$).

• Current Equation:
    - $I = \frac{\epsilon}{R} (1 - e^{-t / \tau})$

• Steady State: When current reaches its maximum, the rate of change is zero, and back emf is zero.

20.7 Energy Stored in a Magnetic Field

• Concept: A battery must do work to establish a current in an inductor because the induced emf opposes the setup.

• Magnetic Energy: The work done is stored as energy in the inductor’s magnetic field.

• Formula:
    - $PE_L = \frac{1}{2} L I^2$