Electromagnetism and Electromagnets

The regenerator, or 'economiser', is Robert Stirling's most significant contribution and the key to the engine's high efficiency. It is an internal heat exchanger that temporarily stores and releases thermal energy during the cycle. As hot gas moves to the cold side, it deposits heat in the regenerator matrix, which is then picked up by the cold gas on its return trip to the hot side.

Engineering stress (\(\sigma\)) is the applied load divided by the original cross-sectional area (\(A_0\)), while engineering strain (\(\epsilon\)) is the change in length (\(\Delta L\)) divided by the original length (\(L_0\)). These definitions, \(\sigma = \frac{F}{A_0}\) and \(\epsilon = \frac{\Delta L}{L_0}\), are fundamental for plotting stress-strain curves but assume the specimen's dimensions remain constant during the test.

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: \(\frac{\partial (\rho \vec{v})}{\partial t} + \nabla \cdot (\rho \vec{v} \otimes \vec{v}) = \nabla \cdot \sigma + \vec{f}\).

The ideal Stirling cycle is a closed regenerative thermodynamic cycle comprising four distinct processes: isothermal expansion, isochoric heat removal, isothermal compression, and isochoric heat addition. The working fluid is permanently contained. Its theoretical thermal efficiency equals the Carnot cycle efficiency, given by \(\eta_{th} = 1 - \frac{T_C}{T_H}\), where \(T_H\) and \(T_C\) are the absolute temperatures of the hot and cold reservoirs.

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.

A magnet's magnetic moment is a vector quantity that characterizes its overall magnetic properties. It can be visualized as a magnetic dipole, a hypothetical pair of north and south poles separated by a distance. The moment points from the south to the north pole. For a current loop, the magnetic moment \(\vec{\mu} = I \vec{A}\), where I is the current and A is the area vector.

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.