Proving Uniform Magnetic Field with Saddle Coils

[Dewgale]

Homework Statement

We have seen that a long solenoid produces a uniform magnetic field directed along the axis of a cylindrical region. However, to produce a uniform magnetic field directed parallel to a diameter of a cylindrical region, one can use saddle coils. The loops are wrapped over a somewhat flattened tube. Assume the straight sections of wire are very long. The overall current distribution is the superposition of two overlapping circular cylinders of uniformly distributed current, one toward you and one away from you. The current density J is the same for each cylinder. The position of the axis of one cylinder is described by a position vector a relative to the other cylinder. Prove that the magnetic field inside the hollow tube is $\frac{\mu_0 J_0}{2}$ downward.

Homework Equations

Ampere's Law

The Attempt at a Solution

Given that
$$\int \vec B \cdot \vec dl = \mu_0 I$$, I set $I = J \ \ A$, where A is the cross sectional area of each cylinder of current.
Then we get $$B*(2 \pi r) = \mu_0 J \pi r^2$$
or
$$B = \frac{\mu_0 J r}{2}$$

Now when I add in the contribution from the other current, which ought to be exactly the same, I get $B = \mu_0 J r$, which is not what I want. I'm not sure how to proceed from here.

Thanks in advance for any help!

 

[TSny]

The two cylinders have currents in opposite directions, so the two B field vectors at a point will tend to cancel. However, they don't cancel exactly because the central axes of the cylinders are displaced by an amount $a$.

 

[Umrao]

The two cylinders have currents in opposite directions, so the two B field vectors at a point will tend to cancel. However, they don't cancel exactly because the central axes of the cylinders are displaced by an amount a.

I tried this question yesterday. But something is pretty off with the description. I tried to google the image of saddle coil as description was way too poor, but the image that I found was way different than what I can make out from question..Dewgale you don't happen to have any image of this question (i.e. an image of saddle coil in your book/paper from where you took the question), do you?

I have attached a figure with the post, to show how I perceive a "part" of the problem - specifically -

The overall current distribution is the superposition of two overlapping circular cylinders of uniformly distributed current, one toward you and one away from you. The current density J is the same for each cylinder. The position of the axis of one cylinder is described by a position vector a relative to the other cylinder.

Considering that the sketched portion of cylinder should be carrying current, while the unsketched is not, area of sketched part comes out by substracting the area of 2 discs with area of ellipse(unsketched part). But that does not give the required answer. Further the unsketched portion may have a constant magnetic field. The current distribution is not completely symmetrical.(consider a point inside the unsketched part of cylinder close to rim and other at middle - current does not quite appear to be symmetrical about these does it?)

The other part with this one is what is confusing -

The loops are wrapped over a somewhat flattened tube. Assume the straight sections of wire are very long.

and

Prove that the magnetic field inside the hollow tube

If I have to consider a coil, a wire - then current density holds very less meaning as generally area is both constant and small. Further if I have to consider that the tube is hollow then my procedure was completely incorrect, but then current density can hold meaning if current flows through the surface of cylinders.

As I said the difficult part of the problem is actually getting the figure straight- Is it what google says, or does it have loops of wire or is there current flowing through the surface of cylinders or does the current flow through the volume of cylinders like in my diagram? Or maybe none of these and current is simply flowing on the moon.

 

[Nidum]

Saddle coils were extensively used as deflector coils for CRT's .
[PLAIN]http://www.indiastudychannel.com/attachments/Resources/161051-12825.jpg[/PLAIN]
http://www.indiastudychannel.com/attachments/Resources/161051-12825.jpg

 

[Umrao]

Thanks to Nidium, now we know - how does it look like - That's the good news- bad news is that it does not look good :-), But let's try

 

[Nidum]

https://books.google.co.uk/books?id=UsX0fq85M2cC&pg=PA114&dq=magnetic+field+in+saddle+shaped+coils&hl=en&sa=X&ved=0ahUKEwjc6cnK3PTKAhWBvhQKHSIkBxIQ6AEIMDAA#v=onepage&q=magnetic field in saddle shaped coils&f=false

There are lots of books and 1950's vintage television magazines with theory of saddle coils but there doesn't seem to be much online .

 

[Umrao]

That was a very misleading question - the link provided by Nidium tells that magnetic field is vertical "at the centre". While a comparison with Solenoid will make it appear as if it will have a constant vertical field

Prove that the magnetic field inside the hollow tube is μ0J02μ0J02\frac{\mu_0 J_0}{2} downward.

 

[TSny]

I have attached a figure with the post, to show how I perceive a "part" of the problem - specifically -

Considering that the sketched portion of cylinder should be carrying current, while the unsketched is not, area of sketched part comes out by substracting the area of 2 discs with area of ellipse(unsketched part). But that does not give the required answer.

It will give the correct answer. Use the superposition principle for magnetic fields. Each entire cylinder produces a magnetic field at some point P in the overlapping region. You do not need to worry about finding the area of the colored regions in the picture attached below.

EDIT: I believe that Dewgale is considering the directions of the currents to be opposite to the directions shown in the picture below. He appears to be considering the case where the net field will be downward.

 

[TSny]

Prove that the magnetic field inside the hollow tube is $\frac{\mu_0 J_0}{2}$ downward.

Did you mean to write $\frac{\mu_0 J_0 \, a}{2}$ ?

 

[Umrao]

Ah! now I get the result of post no. 9. Now I recall that I did a similar question on calculating electric field inside a cavity in an insulator having uniform charge density. Quite similar. But a question, how does this figure look anything like in post 4 or 6.
Edit: I mean - here we are using current flowing through volume of cylinders, while if nothing else - then we should be using loops of wire.

 

[TSny]

But a question, how does this figure look anything like in post 4 or 6.

My interpretation of the figure in post 4 is shown below. You want to concentrate on the currents in the saddle coil wire that run along the straight sections of the coil inside the cylinder. I have tried to indicate these currents by the blue arrows. I have chosen the direction on the left side to run away from us. The current runs in the opposite direction in the wires that are hidden from our view but which I indicated with blue arrows coming toward us. If you imagined looking directly into the cylinder, you would have current distribution similar to the overlapping cylinder current distribution (with current coming toward you on the left and current running away from you on the right).

You should ignore the "toroidal coils" shown in the figure.

 

[Dewgale]

Did you mean to write $\frac{\mu_0 J_0 \, a}{2}$ ?

No, see that's what I got, but according to the question that's incorrect.

I do actually have a picture of the situation:

I've had to hand in the assignment already, but I'm still very curious as to how to make this work.

 

[TSny]

No, see that's what I got, but according to the question that's incorrect.

The statement of the question said that the answer should be $B = \frac{\mu_0 J}{2}$. But this doesn't have the right dimensions to be a magnetic field. It is probably a misprint. I believe the answer is $B = \frac{\mu_0 J a}{2}$.

I'm still very curious as to how to make this work.

For a single cylinder with uniform current density, you were able to show that $B = \frac{\mu_0 J r}{2}$ for points inside the cylinder. Let $\hat{k}$ be a unit vector pointing in the direction of the current density. Verify that the magnitude and direction of $\vec{B}$ are correctly obtained if you write $\vec{B} = -\frac{\mu_0}{2} J \; \vec{r} \times \hat{k}$, where $\vec{r}$ is the position vector locating the field point relative to the axis of the cylinder.

Then consider the superposition of two cylinders with opposite current and axes separated by $\vec{a}$.