Electric Circuits, Oscillations, and Waves: Essential 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.

• Current (I): The rate of flow of electric charge. Defined as $I = \frac{\Delta Q}{\Delta t}$, where $\Delta Q$ is the charge and $\Delta t$ is the time interval.

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

$V = IR$

• Resistance (R): Opposition to current, given by:

$R = \rho \frac{\ell}{A}$ Where $\rho$ is resistivity, $\ell$ is length, and $A$ is cross-sectional area.

Power in Electric Circuits

• Power (P): The rate at which energy is used or dissipated in a circuit:

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

Resistors in Series and Parallel

• Series: Same current flows through all resistors; total resistance is the sum.

$R_{eq} = R_1 + R_2 + R_3$

• Parallel: Same voltage across all resistors; total resistance is given by:

$\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}$

• Current in Series: $I_T = I_1 = I_2 = I_3$

• Voltage in Series: $V_T = V_1 + V_2 + V_3$

• Voltage in Parallel: $V_T = V_1 = V_2 = V_3$

• Current in Parallel: $I_T = I_1 + I_2 + I_3$

Oscillations and Simple Harmonic Motion

Spring Force and Newton's Second Law

• Hooke's Law (Spring Force): The restoring force of a spring is proportional to its displacement:

$F_s = -kx$

• Newton's Second Law: The net force on an object equals mass times acceleration:

$F_{net} = ma$

Frequency, Period, and Angular Frequency

• Frequency (f): Number of oscillations per second.

• Period (T): Time for one complete oscillation: $T = \frac{1}{f}$

• Angular Frequency (\omega): $\omega = 2\pi f$

Energy in Simple Harmonic Motion

• Total Mechanical Energy: The sum of kinetic and potential energy in a mass-spring system:

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

• Velocity as a Function of Position:

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

Equations of Motion for a Mass on a Spring

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

• Angular Frequency: $\omega = 2\pi f = \sqrt{\frac{k}{m}}$

• Displacement: $x(t) = A \cos(2\pi f t)$ or $x(t) = A \sin(2\pi f t)$

• Velocity: $v(t) = -A\omega \sin(2\pi f t)$ or $v(t) = A\omega \cos(2\pi f t)$

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

Waves and Sound

Wave Speed and Linear Density

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

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

• Wave Speed on a String:

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

• Wave Speed in a Medium: For sound in a solid or fluid:

$v_{sound} = \sqrt{\frac{B}{\rho}}$ Where $B$ is the bulk modulus and $\rho$ is the density.

Wave Properties

• Relationship between Speed, Wavelength, and Frequency:

$v = \lambda f$

Intensity and Sound Level

• Intensity (I): Power per unit area:

$Intensity = \frac{Power}{Area}$ For a spherical source, $Area = 4\pi r^2$

• Inverse Square Law for Intensity:

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

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

• Sound Level (\beta) in Decibels (dB):

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

• Intensity from Sound Level:

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

Heat Capacities of Gases (Adiabatic Index)

• Adiabatic Index (\gamma): Ratio of heat capacities for different types of gases:

Type of Gas	Adiabatic Index (\gamma)
Monatomic	1.67
Diatomic	1.4
Triatomic	1.3

Additional info: The adiabatic index is important in thermodynamics and the study of sound propagation in gases.

Summary Table: Key Equations

Concept	Equation
Current	$I = \frac{\Delta Q}{\Delta t}$
Ohm's Law	$V = IR$
Resistance	$R = \rho \frac{\ell}{A}$
Power	$P = IV = I^2R = \frac{V^2}{R}$
Series Resistance	$R_{eq} = R_1 + R_2 + R_3$
Parallel Resistance	$\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}$
Hooke's Law	$F_s = -kx$
Newton's 2nd Law	$F_{net} = ma$
Period (mass-spring)	$T = 2\pi \sqrt{\frac{m}{k}}$
Wave Speed (string)	$v = \sqrt{\frac{T}{\mu}}$
Wave Speed (sound)	$v = \sqrt{\frac{B}{\rho}}$
Intensity	$I = \frac{P}{A}$
Sound Level	$\beta = (10\,dB) \log \left( \frac{I}{I_0} \right)$