User:Jeaucques Quœure/sandbox
The differential rate equation for an elementary reaction using product notation is:
Where:
- is the rate of change of reactant concentration with respect to time.
- k is the rate constant of the reaction.
- represents the concentration of each reactant raised to the power of its stoichiometric coefficient and multiplied together.
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:
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.
The time-dependent Schrödinger equation is given by:
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.
The lateral shift of light passing through a glass slab is given by:
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:
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:
where:
- ti is the thickness of the i-th slab.
- θi is the angle of incidence at the i-th slab.
- θi+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: