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New Equation
the true force between two particles is:
$$\color{#79da2a} \vec{F} = \frac{μ_0}{4\pi} \cdot \frac{q_1 \cdot q_2 \cdot ((\Delta \vec{v}\ \bot\ \hat{r})^2 \- \frac{(\Delta \vec{v}\ \parallel\ \hat{r})^2}{2})}{\vec{r}\ ^2}$$
but we actually want to compare it to the right hand law, so:
taking wire \\(\vec{I_1}\\) and \\(\vec{I_2}\\) lenght super small relative to \\(\vec{r}\\) (basically \\(\vec{I_1} \to d\vec{I_1}\\) and \\(\vec{I_2} \to d\vec{I_2}\\))
The old equation is:
$$ Force\ \vec{I_1}\ makes\ on\ \vec{I_2} = \frac{\frac{μ_0}{4\pi} \cdot (\vec{I_1} \times \hat{r}) \times \vec{I_2} }{\vec{r}\ ^2}$$
$$\color{#f7c860} = \frac{\frac{μ_0}{4\pi} \cdot |I_1| \cdot |I_2| }{\vec{r}\ ^2} \cdot ((\hat{I_1} \times \hat{r}) \times \hat{I_2} )$$
The new equation is:
$$ Force\ \vec{I_1}\ makes\ on\ \vec{I_2} = \frac{\frac{μ_0}{4\pi} \cdot (2 \cdot \vec{I_1} \cdot \vec{I_2} - 3(\vec{I_1} \cdot \hat{r}) \cdot (\vec{I_2} \cdot \hat{r}))}{\vec{r}\ ^2} \cdot \hat{r}$$
$$\color{#f7c860} = \frac{\frac{μ_0}{4\pi} \cdot |I_1| \cdot |I_2| }{\vec{r}\ ^2} \cdot (2 \cdot \hat{I_1} \cdot \hat{I_2} - 3(\hat{I_1} \cdot \hat{r}) \cdot (\hat{I_2} \cdot \hat{r})) \cdot \hat{r}$$
You may say "but zahar it looks so complicated",
bro, maybe complicated but makes soooo much more sense. go over previous links to dive deeper.
After doing integral on closed wire loop, the results of the right hand law and the new equation are the same.
here is a tool that demonstrates their total force results after doing manual integral:
(if you dont trust me you can check the code on GitHub)
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.
[More thoughts here](discussion.html)