equations for heat, electricity, magnetism, and optics

equation for linear thermal expansion, where ΔL is the change in length, α is the coefficient of linear expansion, L is the original length, and ΔT is the change in temperature.

ΔL=αLΔT

formula for the new length after thermal expansion, where Lnew is the final length, Lo is the original length, α is the coefficient of linear expansion, and ΔT is the change in temperature.

L_new=L_o+αL_oΔT

the formula for the area expansion of a two-dimensional object, where Anew is the new area, Lo and Wo are the original length and width, respectively, and α is the coefficient of area expansion. Most cases ignore α²L_oW_oΔT because its so small

A_new=(L_o+αL_oΔT)(W_o+αW_oΔT)=α²L_oW_oΔT+2αL_oW_oΔT+L_oW_o

equation for volumetric thermal expansion, where Vnew is the new volume, Vo is the original volume, α is the coefficient of volumetric expansion, and ΔT is the change in temperature.

V_new=V_o+3αV_oΔT

formula for the change in height due to liquid thermal expansion, where Δhnew is the new height change, ho is the original height, β is the coefficient of linear expansion for height, and ΔT is the change in temperature.

Δh_new=βh_oΔT

represents the formula for thermal energy transfer, where Q is the heat added or removed, m is the mass, c is the specific heat capacity, and ΔT is the change in temperature.

Q=mcΔT

zeroth law of thermodynamics

the state in which all parts of a system have the same temperature, resulting in no net heat flow between them.

first law of thermodynamics, stating that the change in internal energy (ΔU) of a system is equal to the heat added to the system (Q) minus the work done by the system (W).

ΔU=Q+W

ideal gas law, which relates the pressure (P), volume (V), number of moles (N), the ideal gas constant (R), and temperature (T) of an ideal gas.

PV=NRT

equation representing the number of particles in a system, where n is the number of moles and kB is the Boltzmann constant, relating macroscopic and microscopic properties of gases.

N=nK_B

equation for average energy per degree of freedom (L) of an ideal gas in terms of the number of degrees of freedom (Dof), Boltzmann's constant (kB), and the temperature (T), or alternatively, in terms of moles (n) and the ideal gas law.

L=(Dof/2)K_BT=(Dof/2)nRT

equation for average kinetic energy of a particle in a gas, where m is the mass of the particle, v is its velocity, K_B is the Boltzmann constant, and T is the absolute temperature. This relationship illustrates how particle motion relates to thermal energy.

(1/2)mv²=(3/2)K_BT

equation for root mean square velocity of gas particles, where Vrms is the root mean square velocity, K_B is the Boltzmann constant, T is the absolute temperature, and m is the mass of a single particle. This formula relates the speed of particles in a gas to their thermal energy.

V_rms=sqrt(3K_BT/m)

equation representing the ideal gas law, relating pressure (P), volume (V), and temperature (T) of an ideal gas. It shows that for a fixed amount of gas, the ratio of pressure times volume to temperature remains constant.

PV/T=PV/T

equation for heat transfer, where Q is the heat added or removed, n is the number of moles, c is the specific heat capacity, and ΔT is the change in temperature. This equation quantifies the energy required to change the temperature of a substance.

Q=ncΔT

equation for work done by or on a gas during an isobaric process, where W is the work, P is the pressure, and ΔV is the change in volume. This formula indicates that work is done when the volume changes under constant pressure.

W=-PΔV or fnint(dW)=fnint(-PdV)

Monoatomic vs Diatomic Dof

Monoatomic gases have three translational degrees of freedom, while diatomic gases have five degrees of freedom, including rotational.

thermodynamic process where no heat is transferred to or from the system. In an adiabatic process, the change in internal energy is equal to the work done on or by the system, which often results in a change in temperature.

Adiabatic process

principle of conservation of energy in thermodynamics, stating that the total heat exchanged in a system must equal zero, meaning any heat gained by the system must be lost by the surroundings and vice versa.

ΔQ_gain+ΔQ_loss=0

average kinetic energy of a particle in a gas, where K_B represents Boltzmann's constant and T is the temperature in Kelvin.

KE_avg=(3/2)K_BT

equation for average energy per particle in an ideal gas, where Dof is the degrees of freedom, n is the number of particles, R is the ideal gas constant, and T is the temperature in Kelvin.

E=(Dof/2)nRT

types of thermodynamic processes that describe how a system changes while keeping specific variables constant: isometric maintains constant volume, isobaric maintains constant pressure, isothermal maintains constant temperature, and adiabatic has no heat exchange with the surroundings.

Isometric, isobaric, isothermal, adiabatic

describes a thermodynamic process in which no heat is transferred into or out of the system, allowing changes in pressure and temperature to occur solely due to work done on or by the system. ΔQ=0, ΔE=W, W=nc_vΔT, c_v=(Dof/2)R

Adiabatic

equation for multiplicity of a system in statistical mechanics, where Ω represents the number of microstates, q is the number of particles, and N is the number of available states.

Ω=(q+N-1)!/(q!(N-1)!)

equation for entropy of a system in statistical mechanics, where S represents entropy, kB is the Boltzmann constant, and Ω is the multiplicity of microstates. The equation shows that entropy increases with the number of accessible microstates.

S=K_BlnΩ where Ω=Ω₁Ω₂

represents change in entropy (ΔS) of a system, where ΔQ is the heat exchanged and T is the temperature in Kelvin. This formula indicates that the total entropy change is the sum of the entropy changes from different processes (S1 and S2).

ΔS=ΔQ/T and ΔS=S₁+S₂

equation for energy of a quantum harmonic oscillator, where ΔE represents the change in energy, ħ (hbar) is the reduced Planck constant, w is the angular frequency, Ks is the spring constant, and m is the mass of the particle. This expression highlights the relationship between energy and the parameters of the system.

ΔE=h_barw=h_barsqrt(K_s/m)

expression for work done in a thermodynamic process. It calculates the work as the integral of pressure (P) with respect to volume (V), where Pa and Pb are initial and final pressures, and Va and Vb are initial and final volumes.

-fnint(PdV) or (1/2)(P_a+P_b)(V_b-V_a)

thin lens equation, relating the focal length (F) of a lens to the object distance (ID) and the image distance (SD). This equation is fundamental in optics for determining the properties of lenses.

1/F=(1/ID)+(1/SD)

Snell's Law, which describes the relationship between the angles of incidence and refraction when light passes through different media. Here, n1 and n2 are the refractive indices of the respective media.

n₁sin(θ₁) = n₂sin(θ₂)

magnification equation, which relates the height of the image (hi) to the height of the object (ho) and the distances of the image (di) and object (do) from the lens. This equation provides insight into the size and orientation of the image produced by a lens. negative d_i for virtual images, positive for real images

m=(h_i/h_o)=-(d_i/d_o)