连续介质中的动量守恒

电磁铁是一种由电流产生磁场的磁铁。电流切断后，磁场消失。电磁铁通常由一根绕成线圈的导线组成。磁场强度与电流和线圈匝数成正比。这一原理由汉斯·克里斯蒂安·奥斯特发现。

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 [latex]\rho (\frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla \mathbf{v}) = -\nabla p + \mu \nabla^2 \mathbf{v} + \mathbf{f}[/latex].

阴极保护 (CP) 是一种通过将金属表面作为电化学电池的阴极来控制其腐蚀的技术。其原理是通过施加外部电流来抑制金属的自然腐蚀电流。受保护金属的电位被极化至更低的负值，使其进入免疫区或钝化区。

A primary source of electromotive force is electromagnetic induction, described by Faraday's law. It states that a time-varying magnetic flux [latex]\Phi_B[/latex] through a circuit loop induces an EMF ([latex]\mathcal{E}[/latex]). The magnitude of the EMF is proportional to the rate of change of the flux, given by the equation [latex]\mathcal{E} = - \frac{d\Phi_B}{dt}[/latex]. 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 ([latex]\mathcal{E}_{back} \propto \omega[/latex]). 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 [latex]mathcal{E}_L = -L frac{dI}{dt}[/latex], where L is the self-inductance, measured in Henries (H).

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 [latex]\vec{\mu} = I \vec{A}[/latex], where I is the current and A is the area vector.

The Cauchy stress tensor, denoted [latex]\boldsymbol{\sigma}[/latex], 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) [latex]\mathbf{T}[/latex] on any surface passing through that point to the surface's normal vector [latex]\mathbf{n}[/latex] via the linear relationship [latex]\mathbf{T} = \boldsymbol{\sigma} \cdot \mathbf{n}[/latex].

The work-energy theorem states that the net work ([latex]W[/latex]) done by all forces acting on a particle equals the change in its kinetic energy ([latex]\Delta E_k[/latex]). Mathematically, [latex]W = \Delta E_k = E_{k,f} - E_{k,i}[/latex]. This principle directly links the concepts of force, displacement, and energy, providing a powerful tool for analyzing motion without directly using Newton's second law.

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

发电机是一种利用电磁感应将机械能转化为电能的装置。其基本设计包括一个在磁场中旋转的线圈（或一个在线圈内旋转的磁铁）。这种连续旋转会改变穿过线圈的磁通量，从而产生交变电动势 (EMF) 和电流，正如法拉第定律所述。

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.