Understanding Dipole-Loop Interactions in Electrodynamics

[colloio]

Hi Guys

I have hards time doing a question in electrodynamics, i have uploaded the problem and i can't figure out how to do question a), i have hard time understanding what coordinat system i should use, cartesien og cylendrical. I have tryed to draw the field lines, its a bit messy, but its just a dipole above a loop and i have written down the vector potential equation from Grifiths book intro. to electrodynamics.

 

[Adithyan]

Imagine the electric dipole consist of charges +Q and -Q. Since their magnitudes are equal, all the electric field lines originating from +Q terminates at -Q. So apparently, finding flux due to a charge Q at the distance 'd' from the circular wire would suffice.

 

[colloio]

i don't think you have understood my question right

 

[BvU]

We start with a). In the template, you have written down the relevant equation under 2) relevant equations, right ? Small correction to that : $\vec A_{dp}(r)$ should be $\vec A_{dp}(\vec r)$. Which is the vector potential at a point $\vec r$ due to a dipole at the origin. Right ?

So what is needed to rewrite this in the coordinate system of your exercise (which was already chosen for you, so you don't have to worry about that any more!) ?

Oh, and: welcome to PF. Do use the template. It helps you too (in several ways).

 

[colloio]

Yes pretty sure i should use this equation for the question and yes i forgot the arrow on r.

I don't understand "So what is needed to rewrite this in the coordinate system of your exercise (which was already chosen for you, so you don't have to worry about that any more!) ?"

and thank you very much.

Im thinking that i need to use the stated equation to find the vector potential everywhere inside this wire loop, so have to express my r in spherical coordinates and same is for m and than take the crossproduct, and than curl og the vektor potential, which gives me the magentic field and than i can integrate over det area, is this correct?

 

[BvU]

Sort of, yes. Stop worrying about the coordinate system. It is a given. I clearly see an x, a y and a z in the picture.
My "So what is needed ..." tries to point out that this time the dipole is not in the origin, but at (0,0,d)
There is a symmetry around the z axis that allows you to look at (x,0,0) for x from 0 to R and write down $\vec A(x,0,0)$ which I suspect points in the y direction... Your plan of approach seems excellent to me: as you say, take the curl to get B (for which you only need the z component) and integrate.

 

[colloio]

Im pretty weak in vektors for det plan, cylendrical and spherical coordinat systems, but i can see you suggest the plan coordinat system. Well i gave it a try in the sperical coordinate system, i have attached a image of my attempt, but now i don't know what bounderies i should integrate over to get the flux, and don't mind the text, its just the question text in danish language.

 

[Orodruin]

Are you familiar with Stokes' theorem?
$$
\int_S (\nabla \times \vec A(\vec r)) \cdot d\vec S = \oint_{\partial S} \vec A(\vec r) \cdot d\vec r
$$
From this you should be able to perform the flux integral without much problem. You should not even need to compute the actual magnetic field.

 

[colloio]

of coures how could i forget this! great thanks alot. So i actually just need to integrate phi which goes from 0 to 2pi. but what about my r² in the vector potiential dipole, i would need to rewrite that one in terms og θ, right?

 

[Orodruin]

Yes, you need to express $r$ (and $\vec r$) in whatever coordinate system you happen to be using. I would use spherical or cylinder coordinates for this.

 

[colloio]

i have gotten a solution flux = μ_0*m*tanθ/(2*d), can you say anything about this being correct?