Study Notes: LC and LR Circuits, Electromagnetic Induction, and Applications

LC Circuits

Introduction to LC Circuits

An LC circuit consists of a capacitor (with initial charge $Q_0$), an inductor, and a switch. When the switch is closed after being open for a long time, the circuit exhibits oscillatory behavior analogous to a mass-spring system.

• Capacitor: Stores electric charge and energy.

• Inductor: Stores energy in its magnetic field.

• Oscillation: The charge and current oscillate sinusoidally.

Stepwise Behavior of LC Circuits

1. Step A: Capacitor is fully charged ($Q = +Q_0$), current $I = 0$. Analogous to a fully stretched spring (velocity $v = 0$).

2. Step B: Capacitor discharges, current reaches maximum ($Q = 0$, $I = I_{max}$). Analogous to maximum speed of a block on a spring.

3. Step C: Capacitor is recharged with opposite polarity ($Q = -Q_0$, $I = 0$). Analogous to spring compressed in the opposite direction.

4. Step D: Current continues until the initial charge is restored ($Q = 0$, $I = -I_{max}$).

Mathematical Description

• Charge on the capacitor: $Q = Q_0 \\cos(\omega t)$

• Current through the inductor: $I = I_{max} \\sin(\omega t)$

• Angular frequency: $\omega = \frac{1}{\sqrt{LC}}$

Applications of LC Circuits

• Used in radio transmitters and receivers to establish transmission frequency.

• Example: Cell phones use high-frequency LC circuits (e.g., 1000 MHz).

Example Calculation

Given an inductor $L = 1.0$ mH, to achieve a frequency $f = 920$ kHz:

• Use $f = \frac{1}{2\pi\sqrt{LC}}$ to solve for $C$.

LR Circuits

Introduction to LR Circuits

An LR circuit consists of an inductor and a resistor in series, driven by an external battery. When the switch is closed, the current decays exponentially.

• Time constant: $\tau = \frac{L}{R}$

• Exponential decay: $I(t) = I_0 e^{-t/\tau}$

Graphical Representation

• At $t = \tau$, $I$ has decreased to 37% of its initial value.

• At $t = 2\tau$, $I$ has decreased to 13% of its initial value.

QuickCheck Examples

• Immediately after closing the switch, the current is determined by the inductor's opposition to change.

• After a long time, the current is determined by Ohm's law: $I = \frac{V}{R}$.

General Principles of Electromagnetic Induction

Lenz's Law

Lenz's Law states that an induced current in a closed loop occurs only if the magnetic flux through the loop is changing. The direction of the induced current creates a magnetic field that opposes the change in flux.

Faraday's Law

• An emf is induced around a closed loop if the magnetic flux changes.

• Magnitude: $\varepsilon = \left| \frac{d\Phi_B}{dt} \right|$

• Direction: As given by Lenz's law.

Using Electromagnetic Induction

• Model: Make simplifying assumptions.

• Visualize: Use Lenz's law for direction.

• Solve: $\varepsilon = \left| \frac{d\Phi_B}{dt} \right|$ (multiply by $N$ for $N$-turn coil)

• Induced current: $I = \frac{\varepsilon}{R}$

Important Concepts

Magnetic Flux

Magnetic flux ($\Phi_B$) measures the amount of magnetic field passing through a surface:

• $\Phi_B = \vec{A} \cdot \vec{B} = AB \cos \theta$

Changing Magnetic Flux

• Moving a loop into/out of a magnetic field

• Changing the loop's area or orientation

• Changing the magnetic field strength through the loop

Creating Induced Current

• Motional emf: $\varepsilon = v l B$ (due to moving charge carriers in a magnetic field)

• Induced electric field: $\oint \vec{E} \cdot d\vec{s} = -\frac{d\Phi_B}{dt}$ (due to changing magnetic field)

Applications

Inductors

• Solenoid inductance: $L_{solenoid} = \frac{\mu_0 N^2 A}{l}$

• Potential difference: $\Delta V_L = -L \frac{dI}{dt}$

• Energy stored: $U_L = \frac{1}{2} L I^2$

• Magnetic energy density: $u_B = \frac{1}{2\mu_0} B^2$

LC and LR Circuit Summary Table

Circuit Type	Key Equation	Behavior
LC Circuit	$\omega = \frac{1}{\sqrt{LC}}$	Oscillatory (sinusoidal charge and current)
LR Circuit	$\tau = \frac{L}{R}$	Exponential decay of current

Additional info:

• LC circuits are foundational in radio, communications, and signal processing.

• LR circuits are important in transient analysis and filtering applications.