EMF from Faraday’s Law of Induction

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].

الحماية الكاثودية (CP) هي تقنية تُستخدم للتحكم في تآكل سطح المعدن بجعله مهبطًا لخلية كهروكيميائية. يتحقق ذلك بتزويد تيار كهربائي خارجي يُثبط تيارات التآكل الطبيعية. يُستقطب جهد المعدن المحمي إلى قيمة سالبة، مما يُحوّله إلى منطقة مناعة أو سلبية.

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

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).

يحدد قانون لينز اتجاه القوة الدافعة الكهربائية المُستحثة والتيار الناتج عن الحث الكهرومغناطيسي. وينص على أن التيار المُستحث سيتدفق في اتجاه يُنشئ مجالًا مغناطيسيًا يُعاكس التغير في التدفق المغناطيسي الذي أحدثه. هذا المبدأ هو نتيجة لمبدأ حفظ الطاقة، ويُمثل بالإشارة السالبة في قانون فاراداي.

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.

المولد الكهربائي جهاز يُحوّل الطاقة الميكانيكية إلى طاقة كهربائية باستخدام الحث الكهرومغناطيسي. يعتمد تصميمه الأساسي على دوران ملف سلكي داخل مجال مغناطيسي (أو دوران مغناطيس داخل ملف). يُغيّر هذا الدوران المستمر التدفق المغناطيسي المار عبر الملف، مما يُحفّز قوة دافعة كهربائية متناوبة (EMF) وتيارًا كهربائيًا، كما هو موضح في قانون فاراداي.

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.