## Equation that explains everything --- In electricity, many different equations are used to describe the behavior of current, rather than a single, unified model based on electron interactions. For example: - Coulomb's Law is used to calculate the force between two static charges. - Ampère's force law and the Biot-Savart law are needed to determine the force between two current-carrying wires or coils. - Faraday's Law of Electromagnetic Induction describes the voltage generated by a changing magnetic field. ### Unified Equation Instead of relying on these separate theories, I introduce a single model that directly determines the force between charges—and, as a consequence, it naturally produces the exact results of Ampère's force law, Biot-Savart law, and Faraday's Law combined. ### Traditional method issues When using Ampère's force law and the Biot-Savart law to calculate the force between two coils, the positive charges of protons, which remain stationary, are completely ignored. If we account for electrons with charge -q and velocity v, we should also consider protons with charge q and velocity v=0. It's also unclear what reference frame we're using to measure the speed of the electron. What if both coils are moving at different speeds? Furthermore, the "old method" has several issues, such as violating Newton's Third Law, which leads to a breakdown of global momentum conservation. It also disrupts the conservation of angular momentum and violates energy conservation laws, making it mathematically possible, under certain conditions, to generate infinite energy. For example the next situation:
Both particles are positivly charged. The green particle goes right, and the blue particle goes up. blue particle pushes green down, and green particle doesnt creates any force at all on blue. Its nonesense.
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New Equation
I present the potential energy between 2 charges, which automatically ensures that energy and Newton's laws are not violated: $$\color{#e7c472} E_p = q_1 q_2 \frac{\mu_0}{4\pi} \frac{-\frac{1}{2}\Delta v_{r_\Vert}^2 + \Delta v_{r_\bot}^2}{r}$$ When \\(\Delta v_{r_\Vert}\\) is the difference between the charges speed in \\(\hat r\\) (\\(\Delta v \cdot cos(\alpha)\\)), and \\(\Delta v_{r_\bot}\\) is value of \\(\Delta v\\) in \\(r_\bot\\) (\\(\Delta v \cdot sin(\alpha)\\)) Its not something I did in a few days it took me 7 years. For a detailed explanation of how I derived this, check [deep dive](/deep_dive.html) page. From this energy we get that the force is seperate into 2 parts f(r) f(v)... The full explanation is in the [deep dive](/deep_dive.html). shortly: $$\color{#e7c472} E \thickapprox \frac{-v^2}{r}$$ $$\color{#89CFF0} F_{Total} = -\frac{dE}{ds} = \frac{-dE}{dr}\frac{dr}{ds} + \frac{-dE}{dv}\frac{dv}{ds} = \frac{-v^2}{r^2}\cdot 1 + \frac{2v}{r} \cdot \frac{a}{v} =$$ $$\color{#89CFF0}= \frac{-v^2}{r^2} + \frac{2a}{r} = F_r + F_v = F(v) + F(a)$$ so for our E we get: $$\color{#89CFF0} F(v) = q_1 q_2 \frac{\mu_0}{4\pi} \frac{-\frac{1}{2}(\Delta v \cdot \hat r)^2 + (\Delta v \times \hat r)^2}{r^2} \hat r = q_1 q_2 \frac{\mu_0}{4\pi} \frac{-\frac{1}{2}\Delta v_{r_\Vert}^2 + \Delta v_{r_\bot}^2}{r^2} \hat r$$ $$\color{#89CFF0} F(a) = q_1 q_2 \frac{\mu_0}{4\pi} (\frac{\Delta a \cdot \hat r}{r}\hat r - \frac{2 \cdot \hat r \times (\Delta a \times \hat r)}{r}) = q_1 q_2 \frac{\mu_0}{4\pi} \frac{\Delta \vec a_{r_\Vert} - 2 \Delta \vec a_{r_\bot}}{r}$$ But this force is only for charges in cable. Charges that are free to move will have changing \\(\vec v\\) and \\(\vec a\\) so F(a) will also change. If you wanna see force for freely moving electrons (like in plasma) visit [deep dive](/deep_dive.html) ### Methods comparison via manual integration I wrote a code that does manual integration by separating the wires into 1000 parts. The code is available oh [Github](https://github.com/zfrhv/zahar-magnetic-force/blob/main/public/3d_forces.js) and it compares the new vs old methods. The following lines are the forces using ThreeJS: ```js function f_v(dv) { return R_hat.clone().multiplyScalar( (-1/2*dv.dot(R_hat)**2 + dv.clone().cross(R_hat).length()**2) / R.length()**2 ) } function f_a(da) { return R_hat.clone().multiplyScalar( da.dot(R_hat) / R.length() ).add(R_hat.clone().cross(da.clone().cross(R_hat)).multiplyScalar( -2/R.length() )) } ``` But this force is only between 2 charges, and in wire we have both protons and electrons. So we need to calculate the force 4 times: ```js const f_p_n = f_v(v_1_p.clone().sub(v_2_n)).add(f_a(a_1_p.clone().sub(a_2_n))).negate() const f_n_p = f_v(v_1_n.clone().sub(v_2_p)).add(f_a(a_1_n.clone().sub(a_2_p))).negate() const f_n_n = f_v(v_1_n.clone().sub(v_2_n)).add(f_a(a_1_n.clone().sub(a_2_n))) const f_p_p = f_v(v_1_p.clone().sub(v_2_p)).add(f_a(a_1_p.clone().sub(a_2_p))) ``` This is the result of the code (exactly the same results for old and new methods):
Equations comparing (click here for instructions)
symbols: - GREEN WIRE IS THE TOP ONE - BLUE WIRE IS THE BOTTOM ONE - rotating arrow is the rotation force that the wires make on each other - middle arrow is the total force that the circuits make on each other - 4 small aroows on the top circuit are the speed of the loop - orange arrows show the direction of the voltage that the green wire causes on the blue wire usage: - toggle switch is for the force type: off -> the right hand law + Faraday's Law, on: my new equation is used to calculate both forces and the voltage - inputs are for rotation (drag/tap to choose) - the caulculator button is for calculating - you can also rotate the object just by dragging it - you can also zoom in or zoom out with fingers \\ mouse scroll - you can also move in space by grabbing with 2 fingers in the same direction or with keyboard arrows - refresh the page to reset IMPORTANT Notes: - the integral is manual, so of-cource the results wont be 100% accurate - the voltage on the blue wire with my method is calculated when there is some current in that wire. if there is no current in the blue wire then the reults may be different.

These forces that derivated from E are solving all of the problems mentioned at the beginning: - They respect protons changes same as electrons charges - They dont violate energy - They dont violate any momentum - They solve simple current / changing current / wire with speed - They have clear relative point of measurment, we know exatly relative to what the speed is. If the methods are identical in every aspect, why use the new method? Because they only become identical after performing a double integration over closed loops. When comparing the force on a single particle, the methods produce different results, and I believe the new method is more accurate. For example, if you analyze how an electron behaves in plasma or an atom, you may receive different results (more info at [Deep Dive](/deep_dive.html)). Also from the true equation we can learn about the world of charged particles. You can navigate between the pages by clicking on the gear at the top left. contact info: - zahar.pech.13.3@gmail.com - open case on github: [zahar-magnetic-force](https://github.com/zfrhv/zahar-magnetic-force)