Finding Magnetic Field Solutions for a Current-Carrying Cylinder?

[adrian116]

the question is that:

A long, straight, solid cylinder, oriented with its axis in the z-direction, carries a current whose current density is $\vec J$. The current density, although symmetrical about the cylinder axis, is not constant but varies according to the relation

(the relation is in the attachment)

where a is the radius of the cylinder, r is the radial distance from the cylider axis, and $I_0$ is a constant haveing units of amperes.
a) show that $I_0$ is the total current passing through the entire cross section of the wire.
b). Using Ampere's law, derive an expression for the magnitude of the magnetic field $\vec B$ in the region r>=a .
c). Obtain an expression for the current I contained in a circular cross section of radius r<=a and centered at the cylinder axis.
d). Using Ampere's law, derive an expression for the magnitude of the magnetic field $\vec B$ in the region r<=a.


For a, Since for the entire cross section of the wire, i subt. r=a into the relation. But it will give zero. I shown nothing. If I subt. J=I/A,
then $I=2 I_0 [1- (\frac{r}{a})^2]$. Anything wrong,
and how to proof that?

 

[siddharth]

Since the current density is not constant, you need to integrate over the cross section.

Also, have you taken a look at the https://www.physicsforums.com/showthread.php?t=8997"l? If you post the question that way, you won't need to wait till the attachment is approved.

 

[adrian116]

i am sorry since i do not familiar that tutorial yet...

Should i integrate $\frac{2 I_0}{\pi a^2} [1- (\frac{dr}{a})^2]$
from 0 to a? if yes, how to integerate $(dr)^2$

 

[siddharth]

i am sorry since i do not familiar that tutorial yet...

Should i integrate $\frac{2 I_0}{\pi a^2} [1- (\frac{dr}{a})^2]$
from 0 to a? if yes, how to integerate $(dr)^2$

No, that's completely wrong.

If you take a small elemental area da, then the current which flows through that bit is $$\vec{J}.\vec{da}$$

To find the net current through the whole wire, in a sense you add up the current through all the small elemental areas.
So your net current will be

$$I=\int \vec{J}.\vec{da}$$

Now,
(i) Can you tell me what elemental area you will take?
(ii) What will the limits of integration be?

 

[adrian116]

elemental area is the small cross section area $dA=2 \pi r da$,
and the limits of integration is from 0 to a?

 

[adrian116]

I have got the ans.
and the following problems are also be solved,
thank you so much