## Magnetic force equation is WRONG ---

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. Does it really makes sense???



I was trying to find new equation that will make a lot more sense. If you are curious about the proccess check [new equation discovery](discovering_equation.html) (first half is bit boring). If you wanna see the cool features of the new equation, check the [Extreme Cases](different_situations.html) (very interesting). for comparing equations check [Comparing Equations](comparing_equations.html) (its pretty cool).

# 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)

[More thoughts here](discussion.html)