Chapter 9 Physics

9.1 Optics

9.1.1 Snell’s Law

\[ \frac {\sin i}{\sin r} = n \]

• • \(i\): angle of incidence relative to the normal

• • \(r\): angle of refraction relative to the normal

• • \(n\): refractive index of the medium

9.1.2 Lens Formula

\[ \frac {1}{v} + \frac {1}{b} = \frac {1}{f} \]

• • \(v\): image distance

• • \(b\): object distance

• • \(f\): focal length of the lens

9.2 Pressure

9.2.1 Hydrostatic Pressure

\[ p = \rho \cdot g \cdot h + p_0 \]

• • \(p\): pressure at a certain depth

• • \(\rho \): density of the fluid

• • \(g\): gravitational acceleration

• • \(h\): height/depth in the fluid

• • \(p_0\): pressure at the surface (usually atmospheric pressure)

9.2.2 Buoyant Force in a Fluid

\[ F_A = \rho \cdot g \cdot V \]

• • \(F_A\): buoyant force

• • \(\rho \): density of the fluid

• • \(g\): gravitational acceleration

• • \(V\): volume of the displaced fluid

9.3 Gas Laws

9.3.1 Ideal Gas Law

\[ PV = nRT \]

• • \(P\): pressure of the gas

• • \(V\): volume of the gas

• • \(n\): number of moles of gas

• • \(R\): universal gas constant

• • \(T\): temperature in Kelvin

9.3.2 Boyle’s Law

\[ P_1 V_1 = P_2 V_2 \]

• • \(P_1\): initial pressure

• • \(V_1\): initial volume

• • \(P_2\): final pressure

• • \(V_2\): final volume

9.4 Thermodynamics

9.4.1 First Law of Thermodynamics

\[ \Delta U = Q - W \]

• • \(\Delta U\): change in internal energy

• • \(Q\): heat added

• • \(W\): work done by the system

9.4.2 Heat Transfer

\[ Q = mc\Delta T \]

• • \(Q\): heat transferred

• • \(m\): mass of the object

• • \(c\): specific heat capacity of the material

• • \(\Delta T\): change in temperature

9.4.3 Heat Capacity

\[ Q = C \Delta \theta \]

• • \(Q\): amount of heat

• • \(C\): heat capacity of the object

• • \(\Delta \theta \): change in temperature

9.4.4 Heat of Fusion

\[ Q = \ell _s \cdot m \]

• • \(Q\): amount of heat required for melting

• • \(\ell _s\): latent heat of fusion of the material (heat per unit mass)

• • \(m\): mass of the material

9.4.5 Heat of Vaporization

\[ Q = \ell _v \cdot m \]

• • \(Q\): amount of heat required for vaporization

• • \(\ell _v\): latent heat of vaporization of the material (heat per unit mass)

• • \(m\): mass of the material

9.5 Electrostatics

9.5.1 Coulomb’s Law

\[ F = k \frac {|q_1 q_2|}{r^2} \]

• • \(F\): force between the two charges

• • \(q_1\) and \(q_2\): values of the two charges

• • \(r\): distance between the centers of the two charges

• • \(k\): electrostatic constant

• \(k \approx 8.9875 \times 10^9 \, \text {N m}^2/\text {C}^2\)

9.5.2 Electric Field Due to a Point Charge

\[ E = k \frac {|q|}{r^2} \]

• • \(E\): electric field strength at a point

• • \(r\): distance from the charge

• • \(q\): charge

9.5.3 Electric Potential Due to a Point Charge

\[ V = k \frac {q}{r} \]

• • \(V\): electric potential at a point

• • \(r\): distance from the charge

• • \(q\): charge

9.5.4 Relationship Between Electric Field and Potential

\[ E = -\nabla V \]

9.5.5 Capacitance of a Capacitor

\[ C = \frac {\varepsilon _0 \varepsilon _r A}{d} \]

• • \(C\): capacitance

• • \(\varepsilon _0\): vacuum permittivity

• • \(\varepsilon _r\): relative permittivity (dielectric constant) of the material between the plates

• • \(A\): area of one of the plates of the capacitor

• • \(d\): distance between the plates of the capacitor

9.5.6 Energy Stored in a Capacitor

\[ U = \frac {1}{2} C V^2 \]

9.5.7 Continuity Equation (Gauss’ Law for Electricity)

\[ \oint \mathbf {E} \cdot d\mathbf {A} = \frac {Q_{\text {enc}}}{\varepsilon _0} \]

9.6 Electromagnetism

9.6.1 Lorentz force on a moving charge in a magnetic field

\[ F_L = B \cdot |Q| \cdot v \]

• • \(F_L\): Lorentz force

• • \(B\): magnetic field strength

• • \(Q\): charge

• • \(v\): velocity of the charge perpendicular to the magnetic field

9.6.2 Lorentz force on a current-carrying conductor in a magnetic field

\[ F_L = B \cdot I \cdot \ell \]

• • \(F_L\): Lorentz force

• • \(B\): magnetic field strength

• • \(I\): current through the conductor

• • \(\ell \): length of the segment of the conductor in the magnetic field

9.6.3 Magnetic field around a long straight current-carrying wire

\[ B = \frac {\mu _0 \cdot I}{2\pi r} \]

• • \(B\): magnetic field strength at a distance \(r\) from the wire

• • \(\mu _0\): magnetic permeability of vacuum

• • \(I\): current through the wire

• • \(r\): radial distance from the point of interest to the wire

9.6.4 Magnetic field in the core of a long solenoid

\[ B = \frac {\mu _0 \cdot N \cdot I}{\ell } \]

• • \(B\): magnetic field strength in the core of the solenoid

• • \(\mu _0\): magnetic permeability of vacuum

• • \(N\): number of turns of the solenoid

• • \(I\): current through the solenoid

• • \(\ell \): length of the solenoid

9.6.5 Magnetic flux

\[ \Phi = B \cdot A \]

• • \(\Phi \): magnetic flux through a surface

• • \(B\): magnetic field strength perpendicular to the surface

• • \(A\): area through which the magnetic field lines pass

9.6.6 Biot-Savart law

\[ \mathbf {B}(\mathbf {r}) = \frac {\mu _0}{4\pi } \int \frac {I \, d\mathbf {l} \times \hat {r}}{r^2} \]

• • \(\mathbf {B}\): magnetic field at position \(\mathbf {r}\)

• • \(I\): current through the infinitesimally small wire element \(d\mathbf {l}\)

• • \(\hat {r}\): unit vector pointing from the wire element to the observation location

• • \(r\): distance from the wire element to the observation location

• • \(\mu _0\): magnetic permeability of vacuum

9.6.7 Ampère’s law

\[ \oint \mathbf {B} \cdot d\mathbf {l} = \mu _0 I_{\text {enc}} \]

• • \(\mathbf {B}\): magnetic field

• • \(I_{\text {enc}}\): current enclosed by the loop

• • \(\mu _0\): magnetic permeability of vacuum

9.6.8 Faraday’s law of electromagnetic induction

\[ \mathcal {E} = -\frac {d\Phi _B}{dt} \]

• • \(\mathcal {E}\): Induced electromotive force (EMF)

• • \(\Phi _B\): Magnetic flux

9.6.9 Lenz’s law

The direction of the induced current (or EMF) is such that it opposes the change that caused it.

9.6.10 Maxwell’s equations

\begin{align*} \nabla \cdot \mathbf {E} &= \frac {\rho }{\varepsilon _0} \\ \nabla \cdot \mathbf {B} &= 0 \\ \nabla \times \mathbf {E} &= -\frac {\partial \mathbf {B}}{\partial t} \\ \nabla \times \mathbf {B} &= \mu _0 \mathbf {J} + \mu _0 \varepsilon _0 \frac {\partial \mathbf {E}}{\partial t} \end{align*}

• • \(\mathbf {E}\): electric field

• • \(\mathbf {B}\): magnetic field

• • \(\rho \): electric charge density

• • \(\mathbf {J}\): current density

• • \(\varepsilon _0\): electric permittivity of vacuum

• • \(\mu _0\): magnetic permeability of vacuum

9.6.11 Speed of light in vacuum

\begin{align*} c &= \frac {1}{\sqrt {\mu _0 \varepsilon _0}} \\ &\approx 3.00 \times 10^8 \, \text {m/s} \end{align*}

• • \(c\): speed of light in vacuum

• • \(\mu _0\): magnetic permeability of vacuum

• • \(\varepsilon _0\): electric permittivity of vacuum

9.7 Nuclear Physics

9.7.1 Activity of a Radioactive Substance

\[ A(t) = \lambda \cdot N(t) \]

• • \(A(t)\): activity (number of decays per unit of time) at time \(t\)

• • \(\lambda \): decay constant of the substance

• • \(N(t)\): number of undecayed nuclei at time \(t\)

9.7.2 Number of Remaining Nuclei After Time \(t\)

\[ N(t) = N_0 \cdot e^{-\lambda t} \]

• • \(N(t)\): number of undecayed nuclei at time \(t\)

• • \(N_0\): initial number of nuclei at time \(t=0\)

• • \(\lambda \): decay constant of the substance

• • \(t\): time elapsed since the beginning of the observation

9.7.3 Relationship of the Decay Constant with the Half-life

\[ \lambda = \frac {0.693}{T_{1/2}} \]

• • \(\lambda \): decay constant of the substance

• • \(T_{1/2}\): half-life of the radioactive substance (the time in which half of the nuclei decay)

9.8 Kinematics

9.8.1 Displacement with Constant Velocity

\[ x = x_0 + v_x \cdot t \]

• • \(x\): final position at time \(t\)

• • \(x_0\): initial position

• • \(v_x\): velocity in the x-direction (assumed constant)

• • \(t\): elapsed time

9.8.2 Velocity with a Constant Acceleration

\[ v_x = v_{x,0} + a_x \cdot t \]

• • \(v_x\): velocity in the x-direction at time \(t\)

• • \(v_{x,0}\): initial velocity in the x-direction

• • \(a_x\): acceleration in the x-direction (assumed constant)

• • \(t\): elapsed time

9.8.3 Displacement with a Constant Acceleration

\[ x = x_0 + v_{x,0} \cdot t + \frac {a_x \cdot t^2}{2} \]

• • \(x\): final position at time \(t\)

• • \(x_0\): initial position

• • \(v_{x,0}\): initial velocity in the x-direction

• • \(a_x\): acceleration in the x-direction (assumed constant)

• • \(t\): elapsed time

9.9 Dynamics

9.9.1 Newton’s Second Law

\[ F = m \cdot a \]

• • \(F\): net force acting on an object

• • \(m\): mass of the object

• • \(a\): acceleration of the object

9.9.2 Newton’s Law of Universal Gravitation

Gravitational force between two masses:

\[ F = G \cdot \frac {m_1 \cdot m_2}{r^2} \]

• • \(F\): gravitational force between two masses

• • \(G\): gravitational constant

• • \(m_1\) and \(m_2\): masses of the two objects

• • \(r\): distance between the centers of the two masses

9.9.3 Weight of an Object Near the Earth’s Surface

\[ F_z = m \cdot g \]

• • \(F_z\): gravitational force acting on an object

• • \(m\): mass of the object

• • \(g\): acceleration due to gravity (on Earth approximately \(9.81 \, \text {m/s}^2\))

9.9.4 Work Done by a Force

\[ W = F \cdot d \]

• • \(W\): work

• • \(F\): force

• • \(d\): displacement in the direction of the force

9.9.5 Power

\[ P = \frac {W}{t} \]

• • \(P\): power

• • \(W\): work

• • \(t\): time

9.9.6 Spring Force

\[ F_v = k \cdot |\Delta \ell | \]

• • \(F_v\): spring force

• • \(k\): spring constant

• • \(\Delta \ell \): extension or compression of the spring from its equilibrium position

9.9.7 Kinetic Energy

\[ E_k = \frac {m \cdot v^2}{2} \]

• • \(E_k\): kinetic energy

• • \(m\): mass of the object

• • \(v\): velocity of the object

9.9.8 Potential Energy in a Gravitational Field

\[ E_p = m \cdot g \cdot h \]

• • \(E_p\): potential energy

• • \(m\): mass of the object

• • \(g\): acceleration due to gravity

• • \(h\): height above a reference point

9.9.9 Potential Energy in a Spring

\[ E_p = \frac {k \cdot x^2}{2} \]

• • \(E_p\): potential energy of an extended or compressed spring

• • \(k\): spring constant

• • \(x\): extension or compression of the spring from its equilibrium position

9.9.10 Centripetal Acceleration

\[ a_n = \frac {v^2}{r} \]

• • \(a_n\): centripetal acceleration

• • \(v\): velocity of the object

• • \(r\): radius of the circular motion

9.10 Oscillations and Waves

9.10.1 Simple Harmonic Oscillation

\[ y(t) = A \cdot \sin (\omega t + \phi ) \]

• • \(y(t)\): displacement of the object at time \(t\)

• • \(A\): amplitude of the oscillation

• • \(\omega \): angular frequency

• • \(t\): time

• • \(\phi \): phase angle

9.10.2 Wave Equation

\[ y(x,t) = A \cdot \sin (kx - \omega t + \phi ) \]

• • \(y(x,t)\): displacement of the point at position \(x\) at time \(t\)

• • \(A\): amplitude of the wave

• • \(k\): wave number

• • \(\omega \): angular frequency

• • \(x\): position

• • \(t\): time

• • \(\phi \): phase angle

9.10.3 Relation between Angular Frequency and Period

\[ \omega = \frac {2\pi }{T} \]

• • \(\omega \): angular frequency

• • \(T\): period

9.10.4 Relation between Wavelength, Speed, and Frequency

\[ \lambda = \frac {v}{f} = v \cdot T \]

• • \(\lambda \): wavelength

• • \(v\): wave speed

• • \(f\): frequency of the wave

• • \(T\): period of the wave

9.10.5 Period of a Simple Harmonic Oscillation

\[ T = 2\pi \sqrt {\frac {m}{k}} \]

• • \(T\): period of the oscillation

• • \(m\): mass of the oscillating object

• • \(k\): spring constant

9.10.6 Harmonic Frequencies in a Closed Pipe

\[ f_n = n \cdot \frac {v}{4L} \]

• • \(f_n\): \(n\)-th harmonic frequency

• • \(n\): harmonic number (1, 2, 3, ...)

• • \(v\): speed of sound in the pipe

• • \(L\): length of the pipe

9.11 Sound

9.11.1 Speed of Sound in a Medium

\[ v = \sqrt {\frac {B}{\rho }} \]

• • \(v\): speed of sound in the medium

• • \(B\): bulk modulus of the medium

• • \(\rho \): density of the medium

9.11.2 Doppler Effect for a Stationary Observer

\[ f' = f \left ( \frac {c}{c-v_s} \right ) \]

• • \(f'\): observed frequency

• • \(f\): emitted frequency

• • \(c\): speed of sound in the medium

• • \(v_s\): speed of the sound source (positive when approaching the observer, negative when receding)

9.11.3 Sound Intensity

\[ I = \frac {P}{A} \]

• • \(I\): sound intensity

• • \(P\): sound power

• • \(A\): area over which the sound spreads

9.11.4 Sound Level

\[ L = 10 \cdot \log _{10}\left (\frac {I}{I_0}\right ) \]

• • \(L\): sound level in decibels (dB)

• • \(I\): sound intensity

• • \(I_0\): reference intensity (typically \(1 \times 10^{-12} \text {W/m}^2\))

9.12 Physical Constants

9.12.1 Atmospheric Pressure

\[ p_{\text {atm}} = 1.00 \times 10^5 \,\text {Pa} \]

9.12.2 Gravitation

Gravitational Constant

\[ G = 6.67430 \times 10^{-11} \,\text {m}^3\text {kg}^{-1}\text {s}^{-2} \]

Acceleration Due to Gravity near Earth’s Surface

\[ g = 9.81 \, \text {m/s}^2 \]

* Varies depending on location

9.12.3 Speed of Light in Vacuum

\[ c = 2.99792458 \times 10^8 \,\text {m/s} \]

9.12.4 Planck’s Constant

\[ h = 6.62607015 \times 10^{-34} \,\text {J}\cdot \text {s} \]

9.12.5 Gas Constant

\[ R = 8.314462618 \,\text {J}\cdot \text {mol}^{-1}\cdot \text {K}^{-1} \]

9.12.6 Charge of an Electron

\[ e = 1.602176634 \times 10^{-19} \,\text {C} \]

9.12.7 Density

\[ \begin {array}{l|rl} \hline \bf Material & & \bf Density \\ \hline \text {Water at } 20^{\circ }\text {C} & \rho _{\text {water}} = & 998.2 \, \text {kg/m}^3 \\ \text {Lead} & \rho _{\text {lead}} = & 11\,342 \, \text {kg/m}^3 \\ \text {Gold} & \rho _{\text {gold}} = & 19\,320 \, \text {kg/m}^3 \\ \text {Aluminum} & \rho _{\text {aluminum}} = & 2\,700 \, \text {kg/m}^3 \\ \text {Iron} & \rho _{\text {iron}} = & 7\,874 \, \text {kg/m}^3 \\ \hline \end {array} \]

9.12.8 Thermal Properties

\[ \begin {array}{l|rl} \hline \bf Property & & \bf Value \\ \hline \text {Specific heat of water} & c_{\text {water}} = & 4.186 \, \text {J/g} \cdot \text {K} \\ \text {Heat of vaporization of water} & \ell _{\text {v}} = & 2.260 \, \text {kJ/g} \\ \text {Heat of fusion of ice} & \ell _{\text {s}} = & 334 \, \text {J/g} \\ \hline \end {array} \]

9.12.9 Electrostatics

\[ \begin {array}{l|rl} \hline \bf Property & & \bf Value \\ \hline \text {Coulomb's Constant} & k_e = & \frac {1}{4\pi \epsilon _0} \\ \text {Magnetic Permeability of Vacuum} & \mu _0 = & 4\pi \times 10^{-7} \, \text {T} \cdot \text {m/A} \\ \text {Electric Permittivity of Vacuum} & \epsilon _0 = & 8.854 \times 10^{-12} \, \text {C}^2/\text {N} \cdot \text {m}^2 \\ \text {Elementary Charge} & e = & 1.602 \times 10^{-19} \, \text {C} \\ \hline \end {array} \]

9.12.10 Energy

\begin{align*} 1 \, \text {kWh} & = 3.6 \times 10^6 \, \text {J} \\ 1 \, \text {eV} & = 1.602 \times 10^{-19} \, \text {J} \end{align*}

9.12.11 Masses of Fundamental Particles

\[ \begin {array}{l|rl} \hline \bf Particle & & \bf Mass \\ \hline \text {Electron} & m_e = & 9.109 \times 10^{-31} \, \text {kg} \\ \text {Proton} & m_p = & 1.673 \times 10^{-27} \, \text {kg} \\ \text {Neutron} & m_n = & 1.675 \times 10^{-27} \, \text {kg} \\ \hline \end {array} \]

Avogadro’s Number

\[ N_A = 6.022 \times 10^{23} \, \text {mol}^{-1} \]