# Why do these 2 formulas about magnetic field conflict with each other?

$$\newcommand{\tl}[1]{\tag{#1}\label{#1}}$$

Magnetic Field of a current is: $$B = \mu_0 I / (2 \pi r)\tl{01}$$ Magnetic Field of a Moving Charge is: $$B = \mu_0 qv\sin\theta / (4\pi r^2)\tl{02}$$

So I have $$\mu_0 I / (2\pi r) = \mu_0 qv\sin\theta / (4\pi r^2)\implies I = qv\sin\theta / (2r)\tl{03}$$ Since $$I = q/t$$, $$q/t = qv\sin\theta / (2r)\implies vt\sin\theta = 2r\tl{04}$$ Since $$l = vt$$, $$l\sin\theta = 2r\implies\frac{l}{r}\sin\theta = 2\tl{05}$$ Since $$l / r =\cos\theta$$, $$\cos\theta\sin\theta = 2\tl{06}$$

This equation has no solution. It doesn't make sense. Can someone please point out where I'm wrong at?

What I write is just what I think in my head, if you don't understand something, please ask me because you may not find it on any website.

• Why do you think the formula for "a current" is applicable here? Ask yourself: What does the magnetic field of "a current" mean? A circular current? A current through an infinitely long wire? Something else? Commented Dec 28, 2021 at 14:29
• current is an amount of charge passing a point in a wire per second. So I think they are related
– T H
Commented Dec 28, 2021 at 14:48
• The radius $r$ has different geometrical meanings in both formulas for $B$. Therefore it is wrong to equate these two $r$. Commented Dec 28, 2021 at 15:49
• I've edited your question with MathJax. It makes Greek letters easier.
– J.G.
Commented Dec 28, 2021 at 16:12
• Commented Dec 28, 2021 at 22:27