Electromotive Force and Series Connection

A reformulation of classical mechanics based on the principle of stationary action. It uses a scalar quantity called the Lagrangian, defined as kinetic energy minus potential energy (\(L = T - V\)). The equations of motion are derived from the Euler-Lagrange equation, \(\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}_i} \right) - \frac{\partial L}{\partial q_i} = 0\), using generalized coordinates, which simplifies analysis of complex systems with constraints.

Galvanic corrosion is an electrochemical process where one metal corrodes preferentially when in electrical contact with another in the presence of an electrolyte. This occurs because the dissimilar metals create a bimetallic couple, with the less noble (more active) metal acting as the anode and corroding, while the more noble metal acts as the cathode.

The first electric battery, it produces a direct current by stacking pairs of dissimilar metal discs (e.g., zinc and copper) separated by brine-soaked cloth. Each pair forms a galvanic cell, and stacking them in series increases the overall voltage. This arrangement demonstrated the first continuous conversion of chemical energy into electrical energy, paving the way for modern electrochemistry.

The continuum assumption treats fluids as continuous matter rather than as discrete molecules. This simplification is valid when the length scale of the problem is much larger than the intermolecular distance, allowing properties like density and velocity to be defined at infinitesimally small points. This enables the use of differential equations to describe the macroscopic behavior of fluid flow.

The Cauchy stress tensor, denoted \(\boldsymbol{\sigma}\), is a second-order tensor that completely defines the state of stress at a point inside a material. It relates the traction vector (force per unit area) \(\mathbf{T}\) on any surface passing through that point to the surface's normal vector \(\mathbf{n}\) via the linear relationship \(\mathbf{T} = \boldsymbol{\sigma} \cdot \mathbf{n}\).

A primary source of electromotive force is electromagnetic induction, described by Faraday's law. It states that a time-varying magnetic flux \(\Phi_B\) through a circuit loop induces an EMF (\(\mathcal{E}\)). The magnitude of the EMF is proportional to the rate of change of the flux, given by the equation \(\mathcal{E} = - \frac{d\Phi_B}{dt}\). This principle is the foundation for electric generators, transformers, and inductors.

These are two ways to describe motion in continuum mechanics:

• the Lagrangian specification follows individual material particles, tracking their properties over time, like watching a specific car in traffic
• the Eulerian specification focuses on fixed points in space, observing the properties (velocity, density) of whatever particles pass through those points, like a traffic camera observing a fixed intersection.

Solid mechanics is a branch of continuum mechanics that studies the behavior of solid materials, especially their motion and deformation under the action of forces, temperature changes, or other external loads. It is fundamental to engineering for the design and analysis of structures. Key areas include elasticity (recoverable deformation), plasticity (permanent deformation), and fracture mechanics (crack initiation and propagation).

Electromotive force (\(\mathcal{E}\)) is the work done per unit of electric charge by a non-electrical source, such as a battery or generator. Despite its name, it is not a mechanical force but an electric potential, measured in volts. It represents the energy converted from another form (chemical, mechanical) into electrical energy as a charge traverses the source.

The electric current in a Voltaic pile is produced by a redox reaction. At the zinc anode, zinc metal is oxidized, releasing two electrons per atom (\(Zn \rightarrow Zn^{2+} + 2e^{-}\)). These electrons travel through the external circuit to the copper cathode. There, hydrogen ions from the aqueous electrolyte are reduced, forming hydrogen gas (\(2H^{+} + 2e^{-} \rightarrow H_2\)).

A primary flaw of the Voltaic pile is electrode polarization. During operation, hydrogen gas produced at the copper cathode forms an insulating layer of bubbles on its surface. This layer increases the battery's internal resistance and reduces the surface area available for reaction. As a result, the voltage and current output of the pile drop significantly shortly after it begins to be used.

The Navier–Stokes equations are a set of non-linear partial differential equations describing the motion of viscous fluid substances. They are a statement of Newton's second law, balancing momentum changes with pressure gradients, viscous forces, and external forces. For an incompressible fluid, the equation is \(\rho (\frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla \mathbf{v}) = -\nabla p + \mu \nabla^2 \mathbf{v} + \mathbf{f}\).

Cathodic protection (CP) is a technique used to control the corrosion of a metal surface by making it the cathode of an electrochemical cell. This is achieved by supplying an external electrical current that suppresses the natural corrosion currents. The protected metal's potential is polarized to a more negative value, shifting it into a region of immunity or passivity.

This law states that the electromotive force (EMF, \(\mathcal{E}\)) induced in any closed circuit is equal to the negative of the time rate of change of the magnetic flux (\(\Phi_B\)) through the circuit. The mathematical expression is \(\mathcal{E} = -\frac{d\Phi_B}{dt}\). This principle is the basis for electric generators, transformers, and inductors, describing the macroscopic effect of induction.