Magnetic Field Shielding

In summary, the conversation discusses the calculation of magnetic field inside a hollow sphere with a high permeability material, using the method of solving with Laplace's equation. The shielding is found to improve with higher permeability, but the maximum field in the metal also increases. The expression for the magnetic field inside the metal is derived and it is found that it increases as mu gets larger, approaching a constant value of 3 times the initial field. However, it is questioned whether this result is correct and the possibility of overanalysis is considered.
  • #1
king vitamin
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Homework Statement



Consider a uniform [tex]B_{0}[/tex] in the z direction, into which a hollow sphere is placed. If the sphere is made of a material with a high permeability ([tex]B = \mu \ H[/tex]), the magnetic field inside will be greatly reduced. Calculate the magnetic field inside the sphere and the maximum field in the metal.

(my professor talks at length about the method of solving using laplace's eqn., etc)

You will find the shielding improves with [tex]\mu[/tex], but the maximum field in the metal also increases.


The Attempt at a Solution



I've already done all the grunt work - that is, I've calculated the field in all three regions (outside the sphere, inside the sphere, and in the spherical shell). The problem is actually similar to an example in Jackson (page 203), which gives expressions for the field inside and outside which agree with what I obtained, and conclude that the magnetic field inside the sphere decreases as [tex]1/\mu[/tex].

The problem is the second part of the problem - that the maximum magnetic field inside the metal increases with [tex]\mu[/tex]. The expression I obtain for the magnetic field inside the shell (with a being the inner radius, b the outer radius, and [tex]\mu_{r}=\frac{\mu}{\mu_{0}}[/tex]) is as follows:

[tex]\begin{equation}
\frac{3 \ \mu_{r} \ B_{0}}{(\mu_{r}+2)(2\mu_{r}+1) - 2\frac{a^{3}}{b^{3}}(\mu_{r}-1)^{2}} \ \{ ( \ (2 \mu_{r}+1)-\frac{2 a^{3} (\mu_{r}-1)}{r^{3}})cos \theta \mathbf{\hat{r}} - ( \ (2 \mu_{r} + 1) +\frac{a^3(\mu_{r}-1)}{r^{3}})sin \theta \mathbf{\hat{\theta}} \}
\end{equation}
[/tex]

To maximize this, I put in [tex]\theta = 0[/tex] and [tex]r = b[/tex], and the magnitude of the magnetic field (just the r component) is:

[tex]\begin{equation}
\frac{3 \ \mu_{r} \ B_{0} \ \{(2 \mu{r} + 1) - 2 \frac{a^{3}}{b^{3}} (\mu_{r} - 1) \} }{(\mu_{r}+2)(2\mu_{r}+1) - 2\frac{a^{3}}{b^{3}}(\mu_{r}-1)^{2}}
\end{equation}
[/tex]

But as [tex]\mu[/tex] gets large, this doesn't increase, it converges to 3B. Did I not find the maximum of the field in the metal correctly? Am I missing the question somehow? Maybe I took the gradient wrong? (I'm pretty certain I did the "potential" part of the problem correctly)
 
Last edited:
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  • #2
I don't think it is appropriate to pick a value θ = 0 as you have. If I were doing this, I would find the magnitude of the B field first, then see what happens to it as μ gets large. Also, in your expression, are both unit vectors r-hat or should one of them be theta-hat?
 
Last edited:
  • #3
Whoops, the second unit vector should be a theta-hat, I'll edit the post.

If the magnitude is just the r component of the vector, and the r component is proportional to the cosine of theta, then the maximum value would be where cosine is zero, right? Regardless, for some constant theta and r, the limit as mu increases still approaches a constant...

I'm starting to think the expression for the magnetic field is wrong, I'll post the scalar potential from which I derived this tomorrow.
 
  • #4
I've started rethinking this; it doesn't make sense that a metal inside a uniform magnetic field would contain a magnetic field going to infinity inside the metal - I think the question was "show the magnetic field is amplified inside the metal," in which case my answer is correct - as mu gets large, the max field in the metal is 3 times larger than the initial field. I think I just overanalyzed it.
 
  • #5


I would first double check my calculations to ensure that I did not make any errors. Then, I would also consider the limitations of the model being used, such as the assumptions made about the material properties and the geometry of the problem. It is possible that the model may not accurately reflect the behavior of the magnetic field in the real world, and further research or experimentation may be needed to fully understand the phenomenon. Additionally, I would also consider the potential impact of other factors, such as external magnetic fields or temperature, on the shielding effectiveness and maximum field in the metal. Ultimately, the results should be interpreted and communicated with caution, taking into account the uncertainties and limitations of the model.
 

1. What is magnetic field shielding?

Magnetic field shielding is a method used to reduce or block the effect of magnetic fields on a specific area or object. It involves the use of materials with high magnetic permeability to create a barrier that redirects or absorbs the magnetic field lines.

2. Why is magnetic field shielding important?

Magnetic field shielding is important because strong magnetic fields can have harmful effects on electronic devices, sensitive equipment, and even human health. Shielding can protect these objects and individuals from the negative impacts of magnetic fields.

3. What materials are commonly used for magnetic field shielding?

Commonly used materials for magnetic field shielding include mu-metal, iron, steel, nickel, and certain alloys. These materials have high magnetic permeability, which means they are able to redirect and absorb magnetic field lines.

4. How does magnetic field shielding work?

Magnetic field shielding works by creating a barrier between the source of the magnetic field and the protected area or object. The high permeability material in the shielding material attracts and redirects the magnetic field lines, effectively reducing their strength in the protected area.

5. What are the applications of magnetic field shielding?

Magnetic field shielding has a wide range of applications, including in the aerospace industry to protect electronic equipment from the Earth's magnetic field, in medical imaging to reduce interference from outside magnetic fields, and in electronic devices to prevent interference from nearby devices.

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