Magnetic Fields from Currents in a Wire and a Cylindrical Shell

[maiad]

1. Homework Statement [/b]

A solid cylindrical conducting shell of inner radius a = 4.9 cm and outer radius b = 6.1 cm has its axis aligned with the z-axis as shown. It carries a uniformly distributed current I2 = 7.4 A in the positive z-direction. An inifinte conducting wire is located along the z-axis and carries a current I1 = 2.8 A in the negative z-direction.https://www.smartphysics.com/Content/Media/Images/EM/15/h15_cylinders.png

What is ∫B$\bullet$dl where the integral is taken along the dotted path shown in the figure above: first from point P to point R at (x,y) = (0.707d, 0.707d), and then to point S at (x,y) = (0.6d, 0.6d).

Homework Equations

Ampere's Law

The Attempt at a Solution

Not real sure how to start this question. i know the integral =4*pi*10^-7* I(enclosed) are we suppose to find the charge density of the outer circle?

 

[Sunil Simha]

Charge density has no use in this problem. You could start by finding out how much current would be enclosed by that loop if it were complete.

 

[maiad]

Complete as in having a circle having radius d?

 

[Sunil Simha]

Yes that is the loop that I meant. Once you get that, what would be the symmetry of the magnetic field along it? how could you exploit it to get the desired result?

Sorry for replying late. There was apparently some internal server error on PF and I couldn't log in early.

 

[Nickie]

As a scientist, my response to this question would be:

To solve this problem, we can use Ampere's Law which states that the line integral of the magnetic field along a closed path is equal to the product of the enclosed current and the permeability of free space. In this case, we have two sources of current - the cylindrical shell and the infinite wire. We can calculate the enclosed current by finding the net current that passes through the dotted path shown in the figure. This can be done by breaking the path into two parts, from P to R and then from R to S. Using the given current values and the geometry of the problem, we can calculate the net current for each part.

Once we have the enclosed current, we can use Ampere's Law to find the magnetic field at each point along the path. The integral of the magnetic field along the path will give us the total magnetic flux through the path, which is represented by the ∫B\bulletdl term in the question. By solving for this term, we can calculate the magnetic field at each point along the path.

In conclusion, to find the value of ∫B\bulletdl along the dotted path, we need to use Ampere's Law and calculate the enclosed current from both the cylindrical shell and the infinite wire. This will allow us to determine the magnetic field at each point along the path and solve for the desired integral.