Magnetic field decay in cylinder coaxial with solenoid

In summary, the conversation discusses the behavior of a magnetic field and induced electric field in a long cylinder with a superconducting wire and a coaxial cylinder inserted. The question asks about the decay of the magnetic field with distance from the cylinder and the induced E field at the surface of the cylinder. A solution is attempted using an ansatz and approximations, but further analysis is needed to obtain the most general solution and consider the boundary condition at the surface of the cylinder.
  • #1
Alettix
177
11

Homework Statement


For a medium of conductivity ##\sigma##:
$$ \nabla^2 \vec{B} = \sigma \mu \mu_0 \frac{\partial \vec{B}}{\partial t} + \mu \mu_0 \epsilon \epsilon_0 \frac{\partial^2 \vec{B}}{\partial^2 t} $$
A long solenoid with ##r=b## has n turns per unit length of superconducting wire anc carries ##I = I_0 e^{i\omega t} ##. A lonf cylinder of radius a (a<b), permeability ##\mu## and conductivity ##\sigma## is inserted coaxially in the cylinder.

a) If a>>d where ## d = \sqrt{2/\omega \sigma \mu \mu_0}##. Hence describe how the magnetic field decays with distance in from the cylinder.
b) show that the induced E field is:
$$ E = \frac{1}{\sigma d} n I_0 e^{i\omega t } e^{-(a-r)/d} \sin{r/d}$$

Homework Equations


Field in infinite solenoid: ## B = \mu_0 n I##

The Attempt at a Solution


I am unfortunately stuck from the very beginning of this part of the problem. In a long solenoid the field is taken to be entirely along the axis and being uniform. Moreover, ## \nabla \cdot \vec{B} = 0## requires the radial field at the cylinder to be zero in any case and we have cylindrical symmetry. So I suppose what we are looking for is the radial variation of the z component of the field.

I understand that the question is somehow getting to the skin depth, but because we have a second order time derivative this doesn't work out as nicely as with the standard derivation for a perpendicularly incoming electric field.

I have tried to set an ansatz ##B_z = B_0 e^{i(\omega t - kr)} ## and plug into the equation given. Using cylindrical polars this leaves me with $$k^2 + ik/r = \mu \mu_0 (\epsilon \epsilon_0 \omega^2 - \omega \sigma i) $$ which unfortunately doesn't have any nice solution with the skindepth, and worse is dependent on r.

Where do I go wrong and how do I tackle the problem?
Many thanks in advance!

UPDATE: So making some approximations of neglecting curvature and high conductivity, I ended up with: $$ B_z = n I_0 \mu \mu_0 e^{i(\omega t - x/d)} e^{-x/d} $$
ie a traveling wave with quickly decaying amplitude, as expected. (x = a-r)

Is this result correct?

Applying ##\nabla \times \vec{H} \approx \sigma \vec{E} ## it doesn't give me the correct result for the second part. I simply end up with: $$ E =- \frac{1}{\sigma d} n I_0 e^{i(\omega t + x/d) } e^{-(a-r)/d} (1+i)$$
 
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  • #2
Alettix said:
b) show that the induced E field is:
$$ E = \frac{1}{\sigma d} n I_0 e^{i\omega t } e^{-(a-r)/d} \sin{r/d}$$
Should the argument of the sine function be ##(a-r)/d## ?

I have tried to set an ansatz ##B_z = B_0 e^{i(\omega t - kr)} ##
...
...
I ended up with: $$ B_z = n I_0 \mu \mu_0 e^{i(\omega t - x/d)} e^{-x/d} $$
ie a traveling wave with quickly decaying amplitude, as expected. (x = a-r)
Try to obtain the most general solution. You obtained a specific solution representing waves traveling inward (in the positive x-direction) with a decaying factor ##e^{-x/d}##. Is there also a solution representing a wave propagating outward (in the negative x-direction) that contains the same factor ##e^{-x/d}##?

What is the boundary condition for E at the surface of the cylinder? (Hint: superconducting wires)
 

1. How does the magnetic field decay in a cylinder coaxial with a solenoid?

The magnetic field in a cylinder coaxial with a solenoid decays exponentially with the distance from the solenoid. This decay is due to the inverse square law, where the intensity of the magnetic field decreases as the square of the distance from the source.

2. What factors affect the decay of the magnetic field in a cylinder coaxial with a solenoid?

The decay of the magnetic field in a cylinder coaxial with a solenoid is affected by the distance from the solenoid, the strength of the solenoid's magnetic field, and the magnetic permeability of the surrounding material. Additionally, the presence of ferromagnetic materials near the solenoid can alter the decay rate.

3. Can the decay of the magnetic field in a cylinder coaxial with a solenoid be reversed?

No, the decay of the magnetic field in a cylinder coaxial with a solenoid is a natural phenomenon that cannot be reversed. However, the strength of the magnetic field can be increased by adjusting the parameters of the solenoid or by using a stronger magnet.

4. How does the decay of the magnetic field in a cylinder coaxial with a solenoid impact its applications?

The decay of the magnetic field in a cylinder coaxial with a solenoid can limit the range and effectiveness of devices that rely on magnetic fields, such as MRI machines and magnetic sensors. Engineers and scientists must take into account this decay when designing and using such devices.

5. Can the decay of the magnetic field in a cylinder coaxial with a solenoid be measured?

Yes, the decay of the magnetic field in a cylinder coaxial with a solenoid can be measured using a magnetic field strength meter. By taking measurements at different distances from the solenoid, the rate of decay can be determined and used in calculations and experiments.

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