Electric Circuits and Oscillations: Key Equations and Concepts

Electric Circuits

Current, Voltage, and Resistance

Electric circuits involve the flow of electric charge, described by current, voltage, and resistance. Understanding their relationships is fundamental to analyzing circuits.

• Electric Current (I): The rate of flow of electric charge. Defined as:

$I = \frac{\Delta Q}{\Delta t}$

• Ohm's Law: Relates voltage (V), current (I), and resistance (R):

$V = IR$

• Resistance (R): For a uniform conductor, resistance depends on material, length, and cross-sectional area:

$R = \rho \frac{\ell}{A}$

• Power in Electric Circuits: Power dissipated or supplied in a resistor:

$P = IV = I^2R = \frac{V^2}{R}$

Resistors in Series and Parallel

Resistors can be connected in series or parallel, affecting total resistance and current distribution.

• Series Connection:

  • Total resistance: $R_{eq} = R_1 + R_2 + R_3$

  • Current is the same through all resistors: $I_T = I_1 = I_2 = I_3$

  • Total voltage is the sum: $V_T = V_1 + V_2 + V_3$

• Parallel Connection:

  • Reciprocal of total resistance: $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}$

  • Voltage is the same across all resistors: $V_T = V_1 = V_2 = V_3$

  • Total current is the sum: $I_T = I_1 + I_2 + I_3$

Oscillations and Simple Harmonic Motion (SHM)

Spring Force and Newton's Second Law

Oscillatory motion, such as a mass on a spring, is governed by Hooke's Law and Newton's Second Law.

• Hooke's Law (Spring Force):

$F_s = -kx$

• Newton's Second Law:

$F_{net} = ma$

Frequency, Period, and Angular Frequency

Oscillatory systems are characterized by their frequency, period, and angular frequency.

• Frequency (f): Number of cycles per second (Hz):

$f = \frac{1}{T}$

• Period (T): Time for one complete cycle:

$T = \frac{1}{f}$

• Angular Frequency (\omega):

$\omega = 2\pi f$

Energy in SHM

The total mechanical energy in a simple harmonic oscillator is conserved and is the sum of kinetic and potential energies.

• Energy Conservation:

$\frac{1}{2}kA^2 = \frac{1}{2}mv^2 + \frac{1}{2}kx^2$

• Velocity in SHM:

$v = \pm \sqrt{\frac{k}{m}(A^2 - x^2)}$

Equations of Motion for SHM

The position, velocity, and acceleration as functions of time for a mass-spring system:

• Position:

$x(t) = A \cos(2\pi f t)$ $x(t) = A \sin(2\pi f t)$

• Velocity:

$v(t) = -A\omega \sin(2\pi f t)$ $v(t) = A\omega \cos(2\pi f t)$

• Acceleration:

$a(t) = -A\omega^2 \cos(2\pi f t)$ $a(t) = -A\omega^2 \sin(2\pi f t)$

Period and Angular Frequency for a Mass on a Spring

• Period:

$T = 2\pi \sqrt{\frac{m}{k}}$

• Angular Frequency:

$\omega = 2\pi f = \sqrt{\frac{k}{m}}$

Waves and Sound

Wave Speed on a String

The speed of a wave on a stretched string depends on the tension and the linear mass density.

• Linear Density (\mu): Mass per unit length:

$\mu = \frac{m}{L}$

• Wave Speed:

$v_{string} = \sqrt{\frac{T}{\mu}} = \sqrt{\frac{T L}{m}}$

Speed of Sound

The speed of sound in a medium depends on its elastic and inertial properties.

• Speed of Sound:

$v_{sound} = \sqrt{\frac{\gamma k_B T}{m}} = \sqrt{\frac{\gamma R T}{M}}$

• Where \gamma is the adiabatic index (ratio of specific heats):

  • Monatomic: $\gamma = 1.67$

  • Diatomic: $\gamma = 1.4$

  • Triatomic: $\gamma = 1.3$

Wave Properties

• Relationship between speed, wavelength, and frequency:

$v = \lambda f$

Sound Intensity and Decibel Level

Intensity

Sound intensity is the power per unit area. For a spherical source, area is $4\pi r^2$.

• Intensity:

$\text{Intensity} = \frac{\text{Power}}{\text{Area}}$

• Inverse Square Law for Intensity:

$\frac{I_1}{I_2} = \frac{r_2^2}{r_1^2}$

Decibel Scale

The decibel (dB) scale is a logarithmic measure of sound intensity relative to a reference intensity $I_0$.

• Reference Intensity: $I_0 = 1.0 \times 10^{-12} \frac{W}{m^2}$

• Sound Level (\beta):

$\beta = (10\,\text{dB}) \log \left( \frac{I}{I_0} \right)$

• Intensity from Decibel Level:

$I = I_0 10^{\frac{\beta}{10\,\text{dB}}}$

Example: Calculating Total Resistance

• Series: For three resistors of 2 Ω, 3 Ω, and 5 Ω in series:

$R_{eq} = 2 + 3 + 5 = 10\,\Omega$

• Parallel: For the same resistors in parallel:

$\frac{1}{R_{eq}} = \frac{1}{2} + \frac{1}{3} + \frac{1}{5}$

$R_{eq} \approx 0.91\,\Omega$

Example: Period of a Mass-Spring System

• For a 0.5 kg mass and a spring with $k = 200\,N/m$:

$T = 2\pi \sqrt{\frac{0.5}{200}} \approx 0.314\,s$

Example: Sound Intensity at Different Distances

• If $I_1$ is the intensity at $r_1 = 2\,m$, what is the intensity $I_2$ at $r_2 = 4\,m$?

$\frac{I_1}{I_2} = \frac{4^2}{2^2} = \frac{16}{4} = 4$

So, $I_2 = \frac{I_1}{4}$

Table: Comparison of Series and Parallel Resistor Properties

Property	Series	Parallel
Total Resistance	$R_{eq} = R_1 + R_2 + ...$	$\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + ...$
Current	Same through all resistors	Divides among branches
Voltage	Divides among resistors	Same across all resistors

Additional info: Some equations and context were expanded for clarity and completeness, including explicit definitions and examples for exam preparation.