Work-Energy Theorem

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 \(\mathbf{T} = \boldsymbol{\sigma} \cdot \mathbf{n}\).

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.

An electric generator is a device that converts mechanical energy into electrical energy by utilizing electromagnetic induction. The basic design involves a wire coil rotating within a magnetic field (or a magnet rotating within a coil). This continuous rotation alters the magnetic flux passing through the coil, inducing an alternating electromotive force (EMF) and current as described by Faraday's law.

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

An electromagnet is a type of magnet in which the magnetic field is produced by an electric current. The field disappears when the current is turned off. Typically, it consists of a wire wound into a coil. The strength of the magnetic field is proportional to the current and the number of turns in the coil. This principle was discovered by Hans Christian Ørsted.

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.

For continuous systems like fluids or solids, momentum conservation is expressed in a differential form. The rate of change of momentum density \(\rho \vec{v}\) at a point is governed by the divergence of the Cauchy stress tensor \(\sigma\) and body forces \(\vec{f}\). This is described by the Cauchy momentum equation: \(\frac{\partial (\rho \vec{v})}{\partial t} + \nabla \cdot (\rho \vec{v} \otimes \vec{v}) = \nabla \cdot \sigma + \vec{f}\).

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.

As a motor's armature rotates, its conductors cut through the magnetic field, inducing a voltage according to Faraday's law of induction. This 'back-EMF' opposes the main voltage driving the motor. Its magnitude is directly proportional to the motor's rotational speed (\(\mathcal{E}_{back} \propto \omega\)). This phenomenon is crucial for the self-regulation of motor speed and current draw.

Self-inductance is the property of an electrical circuit whereby a change in the current flowing through it induces an electromotive force (EMF) in the circuit itself. This 'back EMF' opposes the change in current, as dictated by Lenz's law. The relationship is given by the formula \(mathcal{E}_L = -L frac{dI}{dt}\), where L is the self-inductance, measured in Henries (H).

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.