Maxwell-Faraday Equation

Lenz's law provides the direction of the induced electromotive force (EMF) and current resulting from electromagnetic induction. It states that the induced current will flow in a direction that creates a magnetic field opposing the change in magnetic flux that produced it. This principle is a consequence of the conservation of energy and is represented by the negative sign in Faraday's law.

A fundamental principle stating that the total energy of an isolated system remains constant over time. Energy can neither be created nor destroyed, only transformed from one form to another, such as from potential to kinetic energy. In classical mechanics, for systems with only conservative forces, the total mechanical energy [latex]E = T + V[/latex] is conserved.

For a Newtonian fluid, viscosity is a function of temperature and pressure but not shear rate. In liquids, viscosity decreases significantly as temperature increases because higher thermal energy allows molecules to overcome cohesive intermolecular forces more easily. Conversely, in gases, viscosity increases with temperature as more frequent molecular collisions at higher speeds lead to greater momentum transfer.

Electromotive force (EMF) and electric potential difference (voltage) are distinct concepts, though both are measured in volts. EMF is the work per unit charge done by a non-conservative force (e.g., chemical reaction, changing magnetic field) to move charge within a source. Potential difference is the work per unit charge done by the conservative electrostatic field between two points.

The Reynolds number ([latex]\text{Re}[/latex]) is a dimensionless quantity in fluid mechanics used to predict flow patterns by representing the ratio of inertial forces to viscous forces. Low Reynolds numbers characterize smooth, orderly laminar flow, while high Reynolds numbers indicate chaotic, eddy-filled turbulent flow. It is crucial for determining the dynamic behavior of a fluid and for scaling experiments.

The Mach number (M) is a dimensionless quantity representing the ratio of flow velocity past a boundary to the local speed of sound: [latex]M = v/a[/latex], where v is the flow velocity and a is the speed of sound. It is the primary indicator of compressibility effects. As Mach number approaches and exceeds 1, the density of the air changes significantly, altering aerodynamic forces.

A reformulation of classical mechanics that uses generalized coordinates and their conjugate momenta. It is based on the Hamiltonian function, [latex]H(q, p, t)[/latex], representing the system's total energy. The dynamics are described by Hamilton's equations: [latex]\dot{q}_i = \frac{\partial H}{\partial p_i}[/latex] and [latex]\dot{p}_i = -\frac{\partial H}{\partial q_i}[/latex]. This framework is central to quantum mechanics and statistical mechanics.

An improvement on the Voltaic pile, the Daniell cell consists of a copper electrode in a copper(II) sulfate solution and a zinc electrode in a zinc sulfate solution, separated by a porous barrier. This two-fluid design prevents hydrogen gas buildup (polarization) on the copper electrode, resulting in a much more stable and reliable voltage source for a longer duration.

Viscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when undergoing deformation. Viscous materials, like honey, resist shear flow and strain linearly with time when a stress is applied. Elastic materials, like a rubber band, strain when stretched and quickly return to their original state once the stress is removed. Viscoelastic materials have elements of both.

The first commercially successful rechargeable battery. It uses a lead (Pb) anode, a lead dioxide (PbO₂) cathode, and a sulfuric acid (H₂SO₄) electrolyte. During discharge, both electrodes are converted into lead sulfate (PbSO₄), and the sulfuric acid is consumed. This process is chemically reversible by applying an external current, making it a practical and robust energy storage system.

The Boltzmann distribution describes the probability that a system in thermal equilibrium at temperature T will be in a specific microstate with energy E. This probability is proportional to the Boltzmann factor, [latex]e^{-E / k_B T}[/latex]. It implies that states with lower energy are exponentially more likely to be occupied than states with higher energy, with temperature modulating this preference.

This foundational formula connects the macroscopic thermodynamic quantity of entropy (S) with the number of possible microscopic arrangements, or microstates (W), corresponding to the system's macroscopic state. The equation, [latex]S = k_B \ln W[/latex], reveals that entropy is a measure of statistical disorder or randomness. The constant [latex]k_B[/latex] is the Boltzmann constant, linking energy at the particle level with temperature.

Shear-thickening, or dilatancy, is a non-Newtonian behavior where viscosity increases with the rate of shear stress. A classic example is a suspension of cornstarch in water (oobleck), which feels liquid when stirred slowly but becomes almost solid when struck or stirred rapidly. This phenomenon is caused by the jamming of suspended particles under high shear.

The Nernst equation relates the reduction potential of a half-cell (or the total voltage of an electrochemical cell) to the standard electrode potential, temperature, and the activities (often approximated by concentrations) of the chemical species undergoing redox. The equation is [latex]E = E^{\circ} - \frac{RT}{nF} \ln Q[/latex], where Q is the reaction quotient.