Given a current, calculate the charge distribution

[fluidistic]

Homework Statement

A ring of radius R has a current density $\vec J=J(r, \theta) \sin \phi \hat \phi$ where phi is the azimuthal angle in spherical coordinates. Calculate the charge distribution considering that it was initially null.

Homework Equations

Not sure. Maybe $\nabla \cdot \vec J + \frac{\partial \rho}{\partial t}=0$.
The divergence theorem.

The Attempt at a Solution

So my idea was to maybe use the continuity equation that I wrote above. From it, I am not sure what to do. Maybe integrate in space so that I can use the divergence theorem, in other words I can reach that $\int _S \vec J \cdot d\vec A + \int \frac{\partial \rho}{\partial t}dV=0$. But I am stuck there because I don't know how to calculate $\vec J \cdot d\vec A$.

Then my other idea is to integrate the continuity equation with respect to time, but again I'm not sure how to do this...
I'd appreciate a little push in the right direction, thanks!

 

[TSny]

I'm not sure what "ring of radius R" means. Maybe it's a ring as shown below. Anyway, I don't think R will play a role.

My guess is that they want you to come up with an expression for $\rho(r, \theta, \phi, t)$. The continuity equation seems like a good approach. What do you get explicitly for $\frac{\partial \rho}{\partial t}$ by evaluating the divergence of $\vec{J}$?

 

[fluidistic]

I'm not sure what "ring of radius R" means. Maybe it's a ring as shown below. Anyway, I don't think R will play a role.

My guess is that they want you to come up with an expression for $\rho(r, \theta, \phi, t)$. The continuity equation seems like a good approach. What do you get explicitly for $\frac{\partial \rho}{\partial t}$ by evaluating the divergence of $\vec{J}$?

Yes that's exactly it and what they ask for.
I took the divergence in spherical coordinates, I reached $\nabla \cdot \vec J = J(r, \theta ) \frac{\cot \theta}{r}=-\frac{\partial \rho}{\partial t}$.
That would make $\rho = - \frac{J(r,\theta) \cot (\theta ) t}{r} + f(r, \theta)$ where f is an arbitrary function appearing when I integrated $\partial \rho$... The result doesn't look right to me, especially this dependence on t, which seems to grow up infinitely.

 

[TSny]

I don't get your expression for the divergence. Make sure to distinguish $\theta$ from $\phi$.

At t = 0 you want $\rho$ to be zero everywhere.

As long as this peculiar current flows, the charge density will grow (positive in some places and negative in others).

 

[fluidistic]

I don't get your expression for the divergence. Make sure to distinguish $\theta$ from $\phi$.

At t = 0 you want $\rho$ to be zero everywhere.

As long as this peculiar current flows, the charge density will grow (positive in some places and negative in others).

\nabla \cdot \vec J ={1 \over r^2}{\partial \left( r^2 J_r \right) \over \partial r}<br /> + {1 \over r\sin\theta}{\partial \over \partial \theta} \left( J_\theta\sin\theta \right)<br /> + {1 \over r\sin\theta}{\partial J_\phi \over \partial \phi} but $J_r=J_\theta=0$ because $\vec J = J_\phi \hat \phi$. I found that divergence formula in https://en.wikipedia.org/wiki/Del_in_cylindrical_and_spherical_coordinates, and the convention used is theta is zenithal while phi is azimuthal, same convention that I use.

 

[TSny]

How do you get a cotangent of theta out of this? Shoudn't the numerator end up with a cosine of phi instead of a cosine of theta?

 

[fluidistic]

How do you get a cotangent of theta out of this? Shoudn't the numerator end up with a cosine of phi instead of a cosine of theta?

My bad, you are correct. I reach $\rho = - J(r, \theta ) \frac{\cos \phi}{r\sin \theta}t$.

 

[TSny]

That looks correct.