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Torodial electric field

Anonymous Sat 31 Jul 15:22:37 2021 No.13455966 View Reply Original Report

Quoted By: >>13455974 >>13456052 >>13456060 >>13458500

Could someone help me solve the maxwell equations for a toroidal coil and/or a toroidal cylinder to calculate the induced electric field around it in case an AC source is connected to the primary loop?

Im trying to figure out the induced electric field strength formula as a function of the radius from the central axis for toroidal coils and toroidal cylinder coils, but I was not able to find anything and Im running out of ideas.

Anonymous

Anonymous Sat 31 Jul 2021 15:26:25 No.13455974 Report

Quoted By: >>13456069

>>13455966
Just write down the expansion in terms of Bessel functions

Anonymous

Anonymous Sat 31 Jul 2021 15:41:16 No.13456001 Report

Quoted By:

Good luck

Anonymous

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Anonymous Sat 31 Jul 2021 15:51:23 No.13456028 Report

Quoted By:

Just do it on your HP bud! Too easy...

Anonymous

Anonymous Sat 31 Jul 2021 16:04:13 No.13456052 Report

Quoted By: >>13456075

>>13455966
$ \begin{align*} \oint\mathbf{B}\cdot d\mathbf{\ell} &= \mu_0\iint\mathbf{J}\cdot d\mathbf{S}\\ B\cdot 2\pi r &= \mu_0 NI\\ B &= \frac{\mu_0NI}{2\pi r} \end{align*} $

Anonymous

Anonymous Sat 31 Jul 2021 16:05:24 No.13456060 Report

Quoted By: >>13456082

>>13455966
>calculate the induced electric field around it in case an AC source is connected to the primary loop?

isn't the solution trivial? the B field is zero. if its AC maxwells equations relate B to E linearly, so if B is zero around E is zero around

Anonymous

Anonymous Sat 31 Jul 2021 16:08:51 No.13456069 Report

Quoted By:

>>13455974

Sorry my knowledge in the field (no pun intended) is a bit rusty. Could you help me with the exact formula/formulas to calculate the induced electric field or a paper providing the exact method on how to solve the maxwell equations for such geometries?

Anonymous

Anonymous Sat 31 Jul 2021 16:10:30 No.13456075 Report

Quoted By: >>13456097

>>13456052
Thats the B field, not the induced E field caused by the changing B field in a toroid coil.

Anonymous

Anonymous Sat 31 Jul 2021 16:14:19 No.13456082 Report

Quoted By: >>13456106

>>13456060
What are you talking about mate? Have you ever connected any type of current to a toroid coil and get a zero B field in the toroid?

Anonymous

Anonymous Sat 31 Jul 2021 16:22:06 No.13456097 Report

Quoted By: >>13456131 >>13456156

>>13456075
$ \begin{align*} \overbrace{\oint \mathbf{E_{ind}}\cdot d\ell}^{e_\text{ind}} &= -\frac{\partial}{\partial t}\overbrace{\iint\mathbf{B}\cdot d\mathbf{S}}^\Phi\\ E_\text{ind}(t)\cdot 2\pi\frac{b - a}{2} &= -\frac{\mu_0N}{2\pi r}\cdot\frac{\partial I(t)}{\partial t}\\ E_\text{ind}(t) &= -\frac{\mu_0 N}{2\pi^2r(b-a)}\cdot\frac{\partial I(t)}{\partial t} \end{align*} $

Anonymous

Anonymous Sat 31 Jul 2021 16:24:27 No.13456106 Report

Quoted By: >>13456118

>>13456082
>What are you talking about mate? Have you ever connected any type of current to a toroid coil and get a zero B field in the toroid?

maybe im reading too much into OPs statement but he says around. in colloquial english that means the region excluding the center. toroidal coils have zero B field everywhere except inside.

Anonymous

Anonymous Sat 31 Jul 2021 16:28:40 No.13456118 Report

Quoted By:

>>13456106
You dont have to have a B field outside the toroid to have an E field outside of it. Based on Faradays law you can define a loop around the closed B field and if that B field changes you have an induced electric field. Thats how and why toroidal transformers work for example.

Anonymous

Anonymous Sat 31 Jul 2021 16:34:39 No.13456131 Report

Quoted By: >>13456156 >>13456249

>>13456097
This looks promising! May I ask what are the variables? Is this the formula for the center of the main axis of the toroid?

Anonymous

Anonymous Sat 31 Jul 2021 16:45:29 No.13456156 Report

Quoted By: >>13456173

>>13456097
>>13456131
It's not promising, it's the induced electromotive force on the current, not the electric field surround the wire. In this expression E isn't even a field, it's just a function of t.

In an ideal conductor case there would be no electric field outside the coil. If there were that would imply storage of electrical energy (capacitive storage). Coils only store magnetic energy (inductive storage).

In reality a wire like that would have some parasitic capacitance to the surrounding environment and corresponding electric field but that's basically impossible to solve analytically.

Anonymous

Anonymous Sat 31 Jul 2021 16:51:00 No.13456173 Report

Quoted By: >>13456199

>>13456156
The question is looking exactly for the induced electric field around the toroid due to a B field change in the toroid. Thats an induced electric field right there I think.

Anonymous

Anonymous Sat 31 Jul 2021 17:01:01 No.13456199 Report

Quoted By: >>13456228

>>13456173
"Around" in the sense of from one end of the wire to the other or "around" in the sense of "in the space surrounding the coil"? These are different things.

Anonymous

Anonymous Sat 31 Jul 2021 17:17:19 No.13456228 Report

Quoted By:

>>13456199
In the space surrounding the coil (starting from the outside and going through the center hole of the toroid then back to the outside).

Anonymous

Anonymous Sat 31 Jul 2021 17:28:52 No.13456249 Report

Quoted By: >>13456261 >>13456406

>>13456131
>May I ask what are the variables?
It's in OP picture.

>Is this the formula for the center of the main axis of the toroid?
It's Faraday's law $\varepsilon = -\frac{\partial\Phi}{\partial t}$ around the toroid when $r >> b > a$ (the toroid is thin) so that magnetic induction is homogeneous inside the toroid.

>>13456131
>It's not promising, it's the induced electromotive force on the current, not the electric field surround the wire
Okay then here's a redo, since I forgot to divide by two and I also completely forgot the surface on the flux part.
$R$ is the distance away from the center of the toroid.

$ \begin{align*} \overbrace{\oint\mathbf{E}\cdot d\ell}^\varepsilon &= -\frac{\partial}{\partial t}\overbrace{\iint \mathbf{B}\cdot d\mathbf{S}}^\Phi\\ E_\text{ind}(t)\cdot 2\pi R &= -\frac{\partial}{\partial t}\left(B(t)\cdot S\right ), && B(t) = \frac{\mu_0 NI(t)}{2\pi r}, && S = \left\{\begin{matrix} R^2\pi, & R < \frac{b - a}{2}\\ \left( \frac{b-a}{2}\right)^2\pi, & R \geq \frac{b - a}{ 2} \end{matrix}\right.\\ \end{align*} $
$ \begin{align*} E_\text{ind}(t) &= \left\{\begin{matrix} -\frac{\mu_0 NR}{4\pi r}\cdot \frac{\partial I(t)}{\partial t},& R < \frac{b - a}{2}\\ -\frac{\mu_0N(b -a )^2}{16\pi rR}\cdot\frac{\partial I(t)}{\partial t}, & R \geq \frac{b -a}{ 2} \end{matrix}\right. \end{align*} $
If the current is AC, then $I(t) = I_m\sin{\omega t}\Rightarrow \frac{\partial I(t)}{\partial t} = I_m\omega\cos(\omega t)$

>In this expression E isn't even a field, it's just a function of t.
It's the intensity of the $\mathbf{E}$ field at distance $R$ away from toroidal center. The field is spinning around the toroid. $\mathbf{E_\text{ind}} = E_\text{ind}(t)\cdot\mathbf{i_R}$

>In an ideal conductor case there would be no electric field outside the coil
There is no magnetic field outside the toroid but the induced electric field exists everywhere.

Anonymous

Anonymous Sat 31 Jul 2021 17:32:48 No.13456261 Report

Quoted By:

>>13456249
Mate if we ever happen to meet IRL and I somehow realize that it was you who answered my question, remind me that I owe you beer. A lot!

Anonymous

Anonymous Sat 31 Jul 2021 18:26:33 No.13456406 Report

Quoted By: >>13458936

>>13456249
Maybe its a stupid question but shouldnt the E fields from different parts of the coil add up in the middle section (hole in the toroid) and subtracted outside of the toroid since in those parts the induced E fields from the opposing side of the toroid with opposing dB/dt vectors are supposed to create opposing E fields as well?

Anonymous

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Anonymous Sat 31 Jul 2021 20:23:16 No.13456761 Report

Quoted By: >>13456914

>plz do my homework for me
lmao at the cucks who fell for it
lmao at OP because he wants to be a scientist, but he can't do math

Anonymous

Anonymous Sat 31 Jul 2021 21:07:11 No.13456914 Report

Quoted By:

>>13456761
Listen up faggit, you do know that "scientists" dont generally understand each and every aspect of science, right? Truly learning the maxwell equations and their meaning as well as their applocations on anything more complex than flat surfaces, squares and simple solenoids takes years if not decades and if you dont use them in your everyday work they fade away rather quickly. GTFO of here and post your shit in some ancient alien thread or something that fits your level of understanding in the field of "science".

Anonymous

Anonymous Sun 01 Aug 2021 08:38:47 No.13458500 Report

Quoted By: >>13458936

>>13455966
im assuming the coil is wound as tightly as a solenoid. Otherwise the integral would be too hard to do. If the radius of the solenoid is much smaller than the radius of the toroid, then we can approx the b field being concentrated in the single ring of radius
>for the central axis for toroidal coils and toroidal cylinder coils
and then use faradays law. Easy peasy

If not tho, you need to do an integral for the mag flux then do faradays law. Not too diff.

Anonymous

Anonymous Sun 01 Aug 2021 14:05:44 No.13458936 Report

Quoted By:

>>13456406
They do. But you try to reach this>>13458500 as much as you can. Even when it's not a toroidal coil, you put the wires in the places those effects are less pronounced

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