Study Notes: Electromagnetic Induction and Inductance (Chapters 29 & 30)

Electromagnetic Induction

Faraday's Law of Induction

Electromagnetic induction refers to the process by which a changing magnetic field induces an electromotive force (emf) in a conductor. This phenomenon is described by Faraday's Law, which is fundamental to understanding how electric generators, transformers, and inductors operate.

• Faraday's Law: The induced emf in a closed loop equals the negative rate of change of magnetic flux through the loop.

$\mathcal{E} = -\frac{d\Phi_B}{dt}$

• Magnetic Flux ($\Phi_B$): The product of the magnetic field, the area it penetrates, and the cosine of the angle between the field and the normal to the area.

$\Phi_B = B A \cos\theta$

• Lenz's Law: The direction of the induced emf (and thus current) opposes the change in magnetic flux that produced it.

Example: Ranking Induced emf from a Graph

• Given a graph of magnetic field magnitude versus time, the induced emf at any instant is proportional to the slope of the graph at that point.

• To rank the magnitude of emf at various points, compare the steepness (rate of change) of the graph at those points.

Additional info: The sign of the emf depends on whether the field is increasing or decreasing, but the magnitude depends only on the absolute value of the slope.

Motional emf and Conducting Loops

When a conductor moves through a magnetic field, an emf is induced due to the motion. This is called motional emf.

• Motional emf: For a straight conductor of length l moving at velocity v perpendicular to a magnetic field B:

$\mathcal{E} = B l v$

• Current in a Moving Loop: If the loop has resistance R, the induced current is:

$I = \frac{B l v}{R}$

• Direction of Induced Current: Determined by Lenz's Law; it opposes the change in flux.

Example: Loop Entering a Magnetic Field

• As a square loop enters a region of uniform magnetic field, the induced current changes direction depending on whether the flux is increasing or decreasing.

• The current is nonzero only while the flux through the loop is changing.

Additional info: The sign convention for current (clockwise or counterclockwise) must be defined for each problem.

Inductance

Mutual Inductance

Mutual inductance quantifies the ability of one coil to induce an emf in another coil due to a changing current.

• Definition: The mutual inductance M between two coils is given by:

$M = \frac{N_2 \Phi_{B2}}{i_1}$

• For two solenoids with N_1 and N_2 turns, cross-sectional area A, and length l:

$M = \mu_0 \frac{N_1 N_2 A}{l}$

• Application: Used in transformers and coupled inductors.

Self-Inductance and RL Circuits

Self-inductance is the property of a coil (or circuit) to oppose changes in current due to the emf induced by its own changing magnetic field.

• Inductance (L): $L = \frac{N \Phi_B}{I}$

• RL Circuit: The current in an RL circuit grows or decays exponentially with time.

$I(t) = I_0 \left(1 - e^{-t/\tau}\right)$, where $\tau = \frac{L}{R}$

• Time Constant ($\tau$): Determines how quickly the current reaches its maximum value.

Example: Comparing RL Circuits

• Given two RL circuits with different inductances, the one with the larger inductance takes longer to reach the same current value.

• If a graph shows both circuits reaching maximum current at the same time, it is incorrect.

Applications and Problem-Solving

Magnetic Damping and Terminal Velocity

When a conducting bar slides on rails in a magnetic field, it experiences a magnetic damping force that opposes its motion. The bar eventually reaches a terminal velocity where the net force is zero.

• Magnetic Force: $F_{mag} = I l B$

• Terminal Velocity: Occurs when the applied force equals the magnetic damping force.

$v_{term} = \frac{F_{app} R}{B^2 l^2}$

• Acceleration as a Function of Velocity:

$a = \frac{F_{app} - \frac{B^2 l^2}{R} v}{m}$

• Velocity as a Function of Time: The velocity increases asymptotically toward the terminal velocity.

Summary Table: Key Equations in Electromagnetic Induction

Concept	Equation	Description
Faraday's Law	$\mathcal{E} = -\frac{d\Phi_B}{dt}$	Induced emf from changing magnetic flux
Motional emf	$\mathcal{E} = B l v$	Emf induced by a moving conductor
Mutual Inductance	$M = \mu_0 \frac{N_1 N_2 A}{l}$	Inductance between two coils
RL Circuit Current	$I(t) = I_0 (1 - e^{-t/\tau})$	Current growth in an RL circuit
Terminal Velocity	$v_{term} = \frac{F_{app} R}{B^2 l^2}$	Final speed of a bar in a magnetic field