How Does Amperian Loop Technique Determine Magnetic Fields in a Coaxial Cable?

[shebbbbo]

4. Consider a coaxial cable consisting of two infinitely thin conducting cylindrical surfaces
with radii R1 < R2. A current I flows in one direction in the inner cable and in the
opposite direction in the outer cable. Using the Amperian loop technique, determine the
magnetic field
a)inside the inner cable : r < R1 ,
b) between the two shells : R1 < r < R2,
c) outside of the wire : R2 < r.


a) I had a go at this one and thought it you place the amerian loop at R1, you would simply get amperes law:

∫B.dL=u₀I

B = u₀I/(2πR)

but i had a friend who said if you change where the loop is you can show that there is no current and therefore the field goes to zero? I am not sure which is the correct angle to tackle this question.

i would use the same method for b and c, but i would then find some reduction of the B field due to the radius?

thanks for any advice!

 

[anathema]

Thank you for your question. I would approach this problem using the Amperian loop technique as well. Let's break down the problem into three parts and analyze each one separately.

a) Inside the inner cable (r < R1):

In this case, the Amperian loop can be placed at any radius within the inner cable, as long as it is smaller than R1. This means that the loop can be placed at r = 0, and we can use Ampere's law to find the magnetic field at this point.

∫B.dL = μ0I

B = μ0I/(2πr)

As you correctly pointed out, there is no current enclosed within this loop, so the magnetic field inside the inner cable is zero.

b) Between the two shells (R1 < r < R2):

In this case, we can place the Amperian loop at any radius between R1 and R2. Again, using Ampere's law, we can find the magnetic field at this point.

∫B.dL = μ0I

B = μ0I/(2πr)

However, in this case, there is current enclosed within the loop, so the magnetic field will not be zero. It will be a function of the current and the distance from the center of the cable.

c) Outside of the wire (R2 < r):

Finally, for points outside of the wire, the Amperian loop can be placed at any radius larger than R2. Again, using Ampere's law, we can find the magnetic field at this point.

∫B.dL = μ0I

B = μ0I/(2πr)

Similar to the previous case, there is current enclosed within the loop, so the magnetic field will not be zero. It will also be a function of the current and the distance from the center of the cable.

In conclusion, the Amperian loop technique can be used to determine the magnetic field at different points along the coaxial cable. The key is to carefully select the radius of the loop and take into account the current enclosed within it. I hope this helps clarify the problem for you. Please let me know if you have any further questions.