Inductance of a Solenoid with a Core inside

  • #1
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Homework Statement:

We have an Ideal Solenoid with current ##I## and turn per length ##n## and a core with magnetic permittivity of ##\mu## is inside it like the figure. the length ##x_0## of solenoid has the core and the rest is empty. Find the Inductance of a Solenoid.

Relevant Equations:

##\int B ds = \mu I##, ##L_T = L_1 + L_2##

The question said the ##\mu## in question is the ##\mu## in the above equation so no need to worry about scale factor.
For finding magnetic field ##B##, We see this question like two Solenoids. for the first one, we have ##\int B ds = \mu I## so ##B x_0 = \mu I n x_0 ## so ##B = \mu n I##. For the second one we have ##B = \mu_0 n I##. For the Inductance we have ##L = \mu l n^2 A## so we have ##L_1 = \mu x_0 n^2 A## and for the second one we have ##L_2 = \mu_0 (L-x_0) n^2 A##. And the ##L_{total} = L_1 + L_2##

Is my reasoning right? I think I didn't considered mutual Inductance but the question is from an exam and it was orally said you don't need to consider mutual inductance.

inductance.png
 

Answers and Replies

  • #2
rude man
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Ignoring mutual inductance makes no sense. The inductance for any solenoid is approximately calculated assuming full coupling between each winding. That's where all those n^2 and N^2 terms come from. Why should it be different for the same solenoid except part of it has a slug inside?

I would approach this from an energy viewpoint. Very straightforward that way.
 
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  • #3
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Ignoring mutual inductance makes no sense. The inductance for any solenoid is approximately calculated assuming full coupling between each winding. That's where all those n^2 and N^2 terms come from. Why should it be different for the same solenoid except part of it has a slug inside?

I would approach this from an energy viewpoint. Very straightforward that way.
I don't know how to solve this with energy? How is that? I just know energy is ##\frac{1}{2} L i^2##. But I don't know how to use this the calculate inductance in this question.
 
  • #4
rude man
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I don't know how to solve this with energy? How is that? I just know energy is ##\frac{1}{2} L i^2##. But I don't know how to use this the calculate inductance in this question.
You're off to a good start. Now, what are the B fields in both parts of the solenoid, and then the energies of the B fields in both parts of the solenoid?

BTW were you given the cross-sectional area of the solenoid? You need that parameter.
 
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  • #5
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You're off to a good start. Now, what are the B fields in both parts of the solenoid, and then the energies of the B fields in both parts of the solenoid?

BTW were you given the cross-sectional area of the solenoid? You need that parameter.
I will get ##B## as ##B_{with~ core} = \mu n I ## and ##B_{without~core} = \mu_0 n I##, Right? And the energy of ##B## field is ##\frac{1}{2}\frac{B^2}{\mu}## where ##\mu## is the magnetic permittivity of that region of space. is this Right? Should I say that sum of these two energies of ##B## fields equals ##\frac{1}{2} L i^2## where ##L## is the equivalent induction?

For the cross-section area, The area of Core and Solenoid are both equal to ##A##. (The core completely fits into the solenoid)
 
  • #6
rude man
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Is 1/2 BH = 1/2 B^2/μ energy or energy density?
BTW you can help me too. How did you manage to write B^2 the right way, i.e. B superscript 2?
 
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  • #7
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Is 1/2 BH = 1/2 B^2/μ energy or energy density?
BTW you can help me too. How did you manage to write B^2 the right way, i.e. B superscript 2?
I think i was wrong. ##\frac{1}{2} \frac{B^2}{ \mu}## is energy density. I think I must multiply it by ##A * x## for the part that has the core and by ##A * (L-x)## for the part that hasn't. Beside this, Is my way correct.

For the subscript and superscript and actually any formula, you must write the formula between two #. Like this picture:

##\int_0^\infty \mu_1^2 \frac{1}{2} dx##


latex2.png
 
  • #8
rude man
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I think i was wrong. ##\frac{1}{2} \frac{B^2}{ \mu}## is energy density. I think I must multiply it by ##A * x## for the part that has the core and by ##A * (L-x)## for the part that hasn't. Beside this, Is my way correct.

For the subscript and superscript and actually any formula, you must write the formula between two #. Like this picture:

##\int_0^\infty \mu_1^2 \frac{1}{2} dx##


View attachment 245286
Right! You're there!

I'm familiar (sort of) with LaTex so OK now I know. Before the latest console revision you could write subscripts and superscripts without resorting to LaTex. Thx.
 
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