User:Jeaucques Quœure/sandbox

The differential rate equation for an elementary reaction using product notation is:

$-{\mathrm {d} \over \mathrm {d} t}[{\text{Reactants}}]=k\prod _{i}[{\text{Reactants}}_{i}]$

Where:

• ${\textstyle -{\mathrm {d} \over \mathrm {d} t}[{\text{Reactants}}]}$ is the rate of change of reactant concentration with respect to time.
• k is the rate constant of the reaction.
• ${\textstyle \prod _{i}[{\text{Reactants}}_{i}]}$ represents the concentration of each reactant raised to the power of its stoichiometric coefficient and multiplied together.

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The second-order differential equation for a serial CLR circuit can be derived from Kirchhoff's voltage law and Ohm's law. It's given by:

$L{\mathrm {d} ^{2}Q \over \mathrm {d} t^{2}}+R{\mathrm {d} Q \over \mathrm {d} t}+{Q \over C}=V(t)$

Where:

• L is the inductance of the circuit.
• R is the resistance of the circuit.
• C is the capacitance of the circuit.
• Q is the charge on the capacitor.
• t is time.
• V(t) is the time-varying voltage source.

This equation describes the behavior of the charge on the capacitor in response to a time-varying voltage source.

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The time-dependent Schrödinger equation is given by:

$\left[U(\mathbf {r} ,t)-i\hbar {\partial \over \partial t}-{\hbar ^{2} \over 2m}\nabla ^{2}\right]\Psi (\mathbf {r} ,t)=0$

Here:

• U(r, t) is the potential energy dependent on position and time.
• i is the imaginary unit.
• ℏ is the reduced Planck's constant.
• m is the mass of the particle.
• ∇² is the Laplacian operator involving spatial derivatives.
• Ψ(r, t) is the wave function dependent on position and time.

This equation describes how the quantum state of a physical system changes with time.

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The lateral shift of light passing through a glass slab is given by:

$d=t\sin(\theta _{1}-\theta _{2})\sec \theta _{2}$

where:

• d is the lateral shift.
• t is the thickness of the slab.
• θ₁ is the angle of incidence.
• θ₂ is the angle of refraction.

Combining this with the Snell's law, the full expression for the lateral shift becomes:

$d=t{\sin \left(\theta _{1}-\arcsin(\sin \theta _{1}\mu _{12})\right) \over \cos \arcsin(\sin \theta _{1}\mu _{12})}.$

The cumulative lateral shift through a combination of glass slabs can be computed with the principle of superposition of individual lateral shifts through each slab. This assumes the angles remain small enough that higher-order effects can be neglected.

The total lateral shift through n slabs is given by:

$D=\sum _{i=1}^{n}t_{i}\sin(\theta _{i}-\theta _{i+1})\sec \theta _{i+1}$

where:

• t_i is the thickness of the i-th slab.
• θ_i is the angle of incidence at the i-th slab.
• θ_i+1 is the angle of refraction for the i-th slab.

Combining this with the Snell's law, the full expression for the total lateral shift through n slabs becomes:

$D=\sum _{i=1}^{n}t_{i}{\sin \left(\theta _{i}-\arcsin(\sin \theta _{i}\mu _{i(i+1)})\right) \over \cos \arcsin(\sin \theta _{i}\mu _{i(i+1)})}.$