We know that when an electron moves at a velocity,it produces magnetic field.
My question is how strong the magnetic field will be if electron moves at a velocity $v$ or with an acceleration $a$ and by what formula.
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For the vectors $,\mathbf E,,\mathbf B,$ of the electromagnetic field produced by an arbitrary moving point charge see equations (01.1),(01.2) in my amswer here : Electric field associated with moving charge, equations (14.14),(14.13) extracted from J.D.Jackson's $^{\prime}$Classical Electrodynamics$^{\prime}$, 3rd Edition. – Frobenius Jan 23 '22 at 07:29
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For the vectors $,\mathbf E,,\mathbf B,$ of the electromagnetic field produced by an uniformly moving point charge see the relativistic equations (01a),(01b) in my amswer here : Magnetic field due to a single moving charge, equations produced by Lienard-Wiechert potentials. In this same answer for the vector $,\mathbf B,$ from a slowly moving point charge see the non-relativistic equation (02) produced by the Biot-Savart Law. – Frobenius Jan 23 '22 at 07:41
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You can use this equation to find the field produced by a moving charge (or an electron in this specific case) with this equation:-
\begin{equation} \mathbf{B}\boldsymbol{=}\dfrac{\mu_{0}}{4\pi}\dfrac{q\left(\boldsymbol{\upsilon}\boldsymbol{\times}\mathbf{\hat{r}}\right)}{r^2} \end{equation} And In the case in which the velocity is time dependent then you will have to substitute the term $v$ with the function of time and you will get the variable magnetic field strength with the variable velocity.
I suggest you to please have a look at this article.
Tejas Dahake
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why mass $m$ is used in the formula? What will be the SI unit of that formula? – Kaushik Kumbhat Jan 23 '22 at 10:37
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@KaushikKumbhat that is not only $m$, I've written $mu$ "mu" is a Greek letter, and here it's used to denote the term know as permeability of air or vacuum, that symbol was not available in my keyboard so I was unable to put the actual symbol. But I'll rectify this issue soon! – Tejas Dahake Jan 23 '22 at 10:59
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You're right. I don't realize, because of the small size font, that you have the unit vector $\mathbf{\hat{r}}$ and not $\mathbf r$. So, I edit your equation to be more clear. If you don't agree I'll turn it as you post it initially.. – Frobenius Jan 23 '22 at 12:26
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@Frobenius No it's completely fine now and looks good than before, thank you brother/sister it actually helped me to understand how to write the answers properly. My mistake that I've written it too small! I apologize for these issues. – Tejas Dahake Jan 23 '22 at 12:31
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Tejas, a note on symbols : for vectors we usually have either boldface symbols like $\mathbf a$, $\boldsymbol \omega$ or regular symbols with an arrow like $\vec{a}$, $\vec{\omega}$ but almost never both of them like $\vec{\mathbf a}$, $\vec{\boldsymbol \omega}$. – Frobenius Jan 23 '22 at 14:06
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