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It would be really nice to see how Jackson got eqn 5.33 on his example problem for finding the vector potential of a circular current loop

$$ J_{\phi}=I\sin\theta'\delta(\cos\theta')\frac{\delta(r'-a)}{a} $$

for describing this geometry:

enter image description here

I've found a few 'explanations' googling, but I'm still confused on the origin of the $\sin\theta$

My best guess

$$ \int \vec{J}\cdot\hat{n} \; d^3x=\int \vec{I}\cdot d\vec{l} \\ \nabla_\phi \left ( \int \vec{J}\cdot\hat{n} \; d^3x=\int \vec{I}\cdot d\vec{l} \right) \\ \frac{1}{\sin\theta} \frac{\partial}{\partial\phi} \int \vec{J}\cdot\hat{n} \; r^2 dr d\cos\theta d\phi = \frac{1}{\sin\theta} \frac{\partial}{\partial\phi} \int I a d\phi \\ \frac{1}{\sin\theta} \int J_\phi r^2 dr d\cos\theta = \frac{1}{\sin\theta} I a \\ \frac{1}{\sin\theta} \int A \delta(\cos\theta) \delta(r-a) r^2 dr d\cos\theta =\frac{1}{\sin\theta} I a \\ Aa^2=Ia \\ A=\frac{I}{a}\\ \therefore J_\phi = I \delta(\cos\theta) \frac{\delta(r-a)}{a} $$

I know the $\sin\theta$ doesn't really matter.. because

$$ \int \sin^2\theta \delta(\theta-\pi/2)d\theta=1 $$

but I'd still like to know where it comes from.

EDIT
The explanation that "it doesn't matter, so lets include it" bothers me. There must be some place in the derivation he picks it up. What possible reason would there be for adding it in at the end?

The change of variables [ref] doesn't work because

$$ \delta(\cos\theta)=\frac{\delta(\theta-\pi/2)}{\sin\frac{\pi}{2}} $$ so you still end up with $$ \int \sin^2\theta \delta(\theta-\pi/2)d\theta $$

I also don't think that it comes out of the Jacobian. Because he later goes on to integrate over the solid angle

$$ \int J r^2 d\Omega dr = \int J r^2 \sin\theta d\theta d\phi $$ which would, again, give something of the form $\sin^2\theta$

EDIT
Jackson's next step (Eqn. 5.35)

$$ A_\phi(r,\theta)=\frac{\mu_0I}{4\pi a}\int r'^2dr'd\Omega' \frac{\sin\theta'\cos\phi'\delta(\cos \theta')\delta(r'-a)}{\left | \mathbf{x}-\mathbf{x'} \right |} \\ \; \\ \text{with } d\Omega' = d\cos\theta' d\phi = \sin\theta' d\theta' d\phi $$

I would agree that a $\sin \theta$ belongs in the formula if he only integrated over $d\theta$ not $\sin\theta d\theta$

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3 Answers 3

up vote 0 down vote accepted

See, you are required to find volume current density $J_\phi$. Though its name is volume current density, you know it is the current flowing per unit surface area. Now the subscript $\phi$ in $J_\phi$ denotes it is flowing in the $\hat{\phi}$ direction. Now in the spherical polar co-ordinate the infinitesimal length elements along the direction $\hat{r},\hat{\theta} \rm\ and\ \hat{\phi}$ are dr, $rd\theta$ and $r\sin\theta d\phi$ respectively. So $dr\times rd\theta$ is the area you are interested in.

Now looking at your figure I am interested in rewriting the current I.

$$I=I\int \delta(r-a) dr= I\int \delta(r-a) dr\ \delta\big(\cos\theta-cos(\pi/2)\big) d(cos\theta)$$ $$=I\int \delta(r-a) dr\ \delta\big(\cos\theta-0\big) d(cos\theta)$$ $$=I\int \delta(r-a) dr\ \delta\big(\cos\theta\big) d(cos\theta)$$ $$=I\int \delta(r-a) dr\ \delta\big(\cos\theta\big) \sin\theta d\theta$$ $$=I\int \delta(r-a)\delta\big(\cos\theta\big) \sin\theta dr\ d\theta$$ $$=I\int \frac{ \delta(r-a)\delta(\cos\theta) \sin\theta}{r} {dr\ r d\theta}$$

So you see your $$J_\phi = I\frac{ \delta(r-a)\delta(\cos\theta) \sin\theta}{r}$$

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Thank you for the reply. Yes, a $\sin\theta$ shows from Jacobian $\sin\theta d\theta$ But, jackson integrates over the full solid angle $d\Omega =\sin\theta d\theta d\phi$ ... which would give a $\sin^2\theta$ I'll edit the question to show his next step –  Ben H. Jul 31 at 16:58
    
I would agree that a $\sin \theta$ belongs in the formula if he only integrated over $d\theta$ not $\sin\theta d\theta$ –  Ben H. Jul 31 at 17:19
    
I think you couldn't follow my answer properly. Take time to read it properly. If it doesn't answer your question completely, point out exactly where you have the problem in my answer. –  user22180 Aug 1 at 5:04
    
Yeah you're right, I wasn't following. I was thinking the $\sin \theta$ was included when the azimuthal angle was held constant (i.e. I thought $dS_\phi=r\sin\theta dr d\theta$ ). I see what you were saying now. thank you –  Ben H. Aug 1 at 5:43
    
So when I did my line integral $\int I \cdot dl$ I should should have evaluated it as $\int I a\sin\theta d\phi$ right? –  Ben H. Aug 1 at 6:26

If you note that $$\delta(\cos\theta)=\frac{\delta(\theta-\pi/2)}{\sin\theta}$$ Then you can see that the sine terms actually cancel out.

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1  
I just want to note that, as I mention in my post, the sine term is completely superfluous. If it weren't, however, there would be no mathematical justification in including it. Your formula should be sin(root), which is 1, not sin(theta). –  Nathan Jul 30 at 13:52
    
I don't think this is correct because the $\sin\theta$ should be $\sin\pi/2$. See edit –  Ben H. Jul 30 at 16:26

EDITED ANSWER: The delta distribution $\delta(x)$ is not unique. It is invariant under transformations of the form $\delta(x) \to f(x)\delta(x)$ where $f(0) = 1$. This is because it is really a distribution and not a function. It is mathematically improper to talk about $\delta(x)$ instead of $\int \delta(x)dx$. Derivations of the term you're interested in will not be unique either. You can show that

$$ I = I\int \delta(r-a)dr = I\int \delta(r-a)\delta(\theta - \pi/2)drd\theta = I\int\frac{\delta(r-a)}{r}\delta(\theta - \pi/2)rdrd\theta. $$

From this expression, it is apparent that we can write the current as

$$ I = \int J_{\phi}rdrd\theta \implies J_\phi = I \frac{\delta(r-a)}{r}\delta(\theta - \pi/2). $$

This result leaves out the $sin(\theta)$ and replaces $a$ with $r$. It really makes no difference because distributions are mathematically represented by $(T, F)$ where $F$ is a set of continuous functions and $T$ is the map from $F \to \mathbb{R}$. While it is common for physicists to represent distributions as just the mapping (i.e., $T = (T,F)$), this is somewhat false and leads to the non-unique representation of quantities like $\delta(x)$.

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I completely agree that it's unnecessary. So why would he put it in? Just to make our lives miserable??? ....what am I saying, that probably is the reason. –  Ben H. Jul 30 at 16:31
    
+1 I get what you were saying now. I'm just thick and needed someone to break it down I guess –  Ben H. Aug 1 at 5:50
    
I tried to do some more research on your question. It seems that everyone has a similar problem and most other textbooks/lectures leave out the sine term. Unless you come across something especially enlightening, I would just accept that it serves no purpose. Also, it doesn't have a physical interpretation either. The natural way to think about current density in this situation is "circulating" about the wire in the xy plane. Why would the angle from the plane influence this expression if it is 0 everywhere but along the circle? –  Nathan Aug 1 at 5:54

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