There are two Laplace operators, the scalar Laplacian and the vector Laplacian. The Laplacian ## \nabla^2 ## used in the equation $$ \nabla \times (\nabla \times \mathbf A) = \nabla (\nabla \cdot \mathbf A) - \nabla^2 \mathbf A $$ is the vector Laplace operator...
Whenever a charged particle is accelerated or decelerated electromagnetic radiation will be produced. The only condition for generating an electromagnetic field in the near-field region and in the far-field region is a charged particle with changing its velocity.
The duty of current balancing is solidarity of both of electricity suppliers and customers. The utilities must do it for voltage balancing by equally distribution of single-phase customers between three phases, while they can do nothing for voltage balancing by equally distribution of...
There are four rules and they are the law of sines, the law of cosines, the law of tangents and the law of cotangents. The law of sines and the law of cosines are more common than the law of tangents and the law of cotangents.
The law of sines and the law of cosines can be used for finding a...
The condition for coplanarity of two lines
## \frac{x-x_1}{a_1} = \frac{y-y_1}{b_1} = \frac{z-z_1}{c_1} ## and ## \frac{x-x_2}{a_2} = \frac{y-y_2}{b_2} = \frac{z-z_2}{c_2} ##
can be written as a triple product and in the next form
## ((x_2-x_1)\hat i + (y_2-y_1)\hat j + (z_2-z_1)\hat k) \cdot...
There is an article (https://en.wikipedia.org/wiki/Polyphase_system) and there is an introduction in the article and there is a sentence in the introduction which claims that compared to a single-phase, two-wire system, a three-phase three-wire system transmits three times as much power for the...
The inverse function $$ F^{-1}(x) $$ of the cumulative density function $$ F(x) = erf(\frac{x}{2a}) – \frac{x}{\sqrt\pi a} exp(-\frac{x^2}{4a^2}) $$ can not be expressed in an analytic form.
It is impossible to solve this problem using only two rules, the junction rule expressed by $$ I_1 + I_2 = I_3 $$ and the rule expressed by $$ v_1 = I_3 R $$ which is called Ohm’s law. Here must be applied one more rule, the loop rule (https://en.wikipedia.org/wiki/Kirchhoff%27s_circuit_laws)...
In Europe when delivering 3 phase power to the panelboard there is the same line-to-neutral voltage on all three phases and that means all three phases together can deliver three times as much power as one of them. In North America when delivering 3 phase power using a high-leg delta there is...
A three-phase system can serve as three single-phase systems and it can be said a three-phase system can deliver three times as much power as a single-phase system.
For high-leg delta three-phase systems this can not be said.
The equation $$ \vec \nabla \cdot \vec B = 0 $$ rules out the existence of magnetic monopoles but there is another equation $$ \vec \nabla \cdot \vec B = \mu _ 0 \rho _ m $$ which includes the existence of magnetic monopoles.
After including time instead of angle in the equations $$ x \left ( \theta \right ) = \left ( R + r \right ) \cos \theta – r \cos \left ( \frac { 3 r + r } { r } \theta \right ) $$ $$ y \left ( \theta \right ) = \left ( R + r \right ) \sin \theta – r \sin \left ( \frac { 3 r + r } { r } \theta...