# Inductance of a Solenoid with a Core inside

## 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.

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rude man
Homework Helper
Gold Member
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.

titansarus
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.

rude man
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Gold Member
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.

titansarus
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)

rude man
Homework Helper
Gold Member
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?

titansarus
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##

rude man
Homework Helper
Gold Member
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.

titansarus