Magnetic pressure on a conductor

[timetraveller123]

Homework Statement

Homework Equations

The Attempt at a Solution

i have done the first two parts but the third part confuses me
i am assuming the current is flowing in the z direction then current is parallel to the magnetic field then why would there be a force acting on it ?[/B]

 

[timetraveller123]

 

[timetraveller123]

i am sorry for the poor quality
a constant 10MA current flows through the long hollow conducting cylinder initially of outer radius of 1cm and inner radius of a . A uniform external magnetic field of 1.0T is applied along the axis of the cylinder.for calculations you may assume the thickness of the cylinder is negligible
qn) calculate the force acting on a thin shell of current r and r + Δr where Δr << b-a ?

once again sorry

i am thinking there is no force as the current is parallel to the magnetic field

 

[rude man]

A constant 10mA current flows through the long hollow conducting cylinder initially of outer radius b and inner radius a . A uniform external magnetic field of 1.0T is applied along the axis of the cylinder.for calculations.

"You may assume the thickness of the cylinder is negligible".

I don't understand why this statement is made. The thickness is b - a.

qn) calculate the force acting on a thin shell of current r and r + Δr where Δr << b-a ?
i am thinking there is no force as the current is parallel to the magnetic field

In the region a < r < b there is a second B field, one set up by the current, . As shown in your illustration, this field is in the θ direction. The ensuing force will be in the r direction and so will tend to expand the outer radius, which is why the problem states "initially of outer radius b".

I have made slight corrections to your posted text.
You're right, the externally applied B field is in the z direction and so exerts no force on the cylinder.
You probably also need to modify your answer to (b) ii.

 

[timetraveller123]

@rude man
thanks for replying it really means a lot to me and thanks for your interest
i have several doubts

1)

In the region a < r < b there is a second B field, one set up by the current, . As shown in your illustration, this field is in the θ direction.

i am not really too familiar with cylindrical coordinates but if a current were to set up a magnetic field would it not be curling around it and not in the theta direction as shown in the diagram in this problem the theta is perpendicular to r unit vector and the wire ie magnetic field is originating from the wire which i thought is not possible am i missing something please help me clarify?

2)

The ensuing force will be in the r direction and so will tend to expand the outer radius, which is why the problem states "initially of outer radius b".

this one confuses me too because i have been brought to believe that something is not affected by the fields it produces in this case the magnetic field does not affect the wire

please help me clarify thanks once again

 

[rude man]

@rude man
i am not really too familiar with cylindrical coordinates but if a current were to set up a magnetic field would it not be curling around it and not in the theta direction as shown in the diagram in this problem the theta is perpendicular to r unit vector and the wire ie magnetic field is originating from the wire which i thought is not possible am i missing something please help me clarify?

"Curling around it' is in the theta direction! Just apply Ampere's law to a B loop within a < r < b.

this one confuses me too because i have been brought to believe that something is not affected by the fields it produces

That's a very general statement, and incorrect. It's true for example that a charge is not affected by its own E field alone but certainly a current can produce its own B field, including within the current that causes it! And so the Lorentz force can exert a force on the shell.

 

[timetraveller123]

"Curling around it' is in the theta direction! Just apply Ampere's law to a B loop within a < r < b.

sorry got confused by the diagram

That's a very general statement, and incorrect. It's true for example that a charge is not affected by its own E field alone but certainly a current can produce its own B field, including within the current that causes it! And so the Lorentz force can exert a force on the shell.

thanks for the wonderful hint really helped me to see what is going on

I don't understand why this statement is made. The thickness is b - a.

now i know why it is because in this competition the use of integrals is not needed but can be used
with that being said i proceed by using integrals

by considering the cross sectional area of the cylinder
a ring of internal radius r and external radius of r + Δr
magnetic field in the ring
L - length of cylinder
J - current density
$B = \frac{J \pi(r^2 - a^2) \mu_o}{2 \pi r}\\ I = J 2 \pi r dr\\ df= I L B\\ df = L \frac{J \pi(r^2 - a^2) \mu_0}{2 \pi r}J 2 \pi r dr\\ df = L \mu_0 J^2 \pi(r^2 - a^2) dr$
is this the answer for the question
is the force radially outward
assuming it is correct (correct me if am wrong)

moving on to the next part of the question about calculating the pressure on curved walls$\int df = \int _{r = a} ^b L \mu_0 J^2 \pi(r^2 - a^2) dr\\ F = L \mu_0 J^2 \pi \int _{r = a} ^b (r^2 - a^2) dr\\ F = L \mu_0 J^2 \pi [\frac{r^3}{3} - a^2 r]_a ^b\\ P = \frac{L \mu_0 J^2 \pi [\frac{r^3}{3} - a^2 r]_a ^b}{2 \pi b L}\\$
how am i supposed to get a muerical answer out of this help again?

edit:
or for pressure on curved surface am i only supposed to use the force on the outermost shell

 

[rude man]

by considering the cross sectional area of the cylinder
a ring of internal radius r and external radius of r + Δr
magnetic field in the ring
L - length of cylinder
J - current density
$B = \frac{J \pi(r^2 - a^2) \mu_o}{2 \pi r}\\ dI = J 2 \pi r dr\\ df= dI L B\\ df = L \frac{J \pi(r^2 - a^2) \mu_0}{2 \pi r}J 2 \pi r dr\\ df = L \mu_0 J^2 \pi(r^2 - a^2) dr$
is this the answer for the question

the question is a bit nebulous IMO so I would say it's OK.

is the force radially outward

If you set up the coordinate system the way I described,
dI = dI k
so B = B θ
and dF = I (dl x B)
which tells you the direction of dF.
Or you can pick a point. say x = r, y = 0 where B is in the +j direction so that k x j = -i
which come to think of it is radially inward and so will tend to contract the shell's radius rather than expand it as I told you earlier. Sorry!

moving on to the next part of the question about calculating the pressure on curved walls
$\int df = \int _{r = a} ^b L \mu_0 J^2 \pi(r^2 - a^2) dr\\ F = L \mu_0 J^2 \pi \int _{r = a} ^b (r^2 - a^2) dr\\ F = L \mu_0 J^2 \pi [\frac{r^3}{3} - a^2 r]_a ^b\\ P = \frac{L \mu_0 J^2 \pi [\frac{r^3}{3} - a^2 r]_a ^b}{2 \pi b L}\\$

This looks good.

how am i supposed to get a muerical answer out of this help again?
edit:
or for pressure on curved surface am i only supposed to use the force on the outermost shell

Why can't you numerically evaluate your expression for P? You have all the numbers you need.
The force is exerted on all the current form a to b. At a the current is zero and at b it is max. When you integrated you got all the force on all the dr current elements.

 

[timetraveller123]

Why can't you numerically evaluate your expression for P? You have all the numbers you need.

i don't have a

 

[rude man]

i don't have a

You're right, you don't, so leave the answer with a in it.

 

[rude man]

I was thinking that since the instructor said calculus was not needed, you could go as follows: the cylinder is very thin so the current I flows in a thinshell at r=b=1 cm. Then run your ampere's law contour just above r = b so the B field there is simply μ₀I/2πb and the force would be F = IL x B directed into the cylinder, with the pressure p = F/2πLb. This isn't rigorous & may even be wrong but it would avoid calculus. Hard to decide what exactly the problem authors had in mind.

 

[timetraveller123]

sorry for the very late reply had a lot of work

is simply μ0I/2πb and the

i believe this is the total current through the conductor
a

force would be F = IL x B

and what is the current here how to calculate that

directed into the cylinder,

and why is it inside all along i have assuming the force was acting outwards

and sorry i am not really getting your method it would be very helpful if you could clarify it for me thanks

 

[rude man]

and what is the current here how to calculate that

Total current as given: 10 mA.

and why is it inside all along i have assuming the force was acting outwards

Pick any point along the contour r. Let's pick x=+r, y=0, z anywhere 0<z<L. Current in the +z direction so B is in the +θ direction which at that point is the +y direction. So at that point,
F = iL x B
L = L k
B = B j
k x j = -i
in other words, inward.
You can pick any other point along the contour, the result will always be the same: the force is inward. The effect is to squeeze the cylindrical shell inwards all around the circumference.

 

[timetraveller123]

ok now i understand the direction and calculation yielded 32 giga pa which is one of the options but i still don't get the method

the magnetic field is caused by all of the current i suppose
then in why is it that in the expression F = I L B
I is also the total current is it because the cylinder is thin and that it can just be approximated as a thin wire

 

[timetraveller123]

um sir is the answer correct
nextly
the next part of the question asks
the cylinder is flexible and allowed to move
explain what happens to the area magnetic field and magnetic flux within the cylinder as the cylinder moves(eg. contracts or expands ) as time passes

 

[rude man]

ok now i understand the direction and calculation yielded 32 giga pa which is one of the options but i still don't get the method

the magnetic field is caused by all of the current i suppose
then in why is it that in the expression F = I L B
I is also the total current is it because the cylinder is thin and that it can just be approximated as a thin wire

You can think of the cylindrical shell as comprising a large number of parallel wires. Each thin wire carries a small portion ΔI of the total current I and is affected by the magnetic field due to all the remaining thin wires. So for each wire there is a small force BL ΔI. Adding all the wires' forces gives you the total force.

 

[rude man]

um sir is the answer correct
nextly
the next part of the question asks
the cylinder is flexible and allowed to move
explain what happens to the area magnetic field and magnetic flux within the cylinder as the cylinder moves(eg. contracts or expands ) as time passes

Just apply Ampere's law to whatever the shell changes to. I would assume finite thickness (b - a) to visualize this better.

 

[timetraveller123]

You can think of the cylindrical shell as comprising a large number of parallel wires. Each thin wire carries a small portion ΔI of the total current I and is affected by the magnetic field due to all the remaining thin wires. So for each wire there is a small force BL ΔI. Adding all the wires' forces gives you the total force.

that was the original integration method but in the new method we seem to calculate the magnetic field of all the small wires as a whole and calculate the effect of magnetic field on all of the wires that is once again like a thin wire being affected by its own magnetic field
the reason why it strikes me odd u asked me to take the magnetic field at r = just more than b ie at that since we are outside of the wire all of the current contributre to the magnetic field and then you asked me to use that to calculate the force on the wire which seems like treating the magnetic field as an external one which is what striked me odd
anyway this question seems odd to me also without integration

 

[timetraveller123]

sorry for the late reply have been busy with academic matter exams are coming hope you understand thanks

Just apply Ampere's law to whatever the shell changes to. I would assume finite thickness (b - a) to visualize this better.

well the area just increases or decreases accordingly
the magnetic field through the hollow part of the cylinder remains the same while the flux would change accordingly as well
what is going on here this just seems to me weird

the external magnetic field in the question doesn't seem to serve any purpose
why would increasing the area have any effect please explain

 

[rude man]

Well, as I said, the only good way to see this is to assume a finite thickness (b - a). Then the force pulling the cylinder inwards is zero at a and max. at b and yes to get the right answer you have to integrate this force from a to b.

Maybe they wanted you to take the mid-point of this force, I don't know. If you assume an arbitrarily thin wall I agree it's not obvious how a force is generated on the cylinder wall. I'm not sure where you'd put your contour to invoke Ampere's law. With a finite thickness it's obvious.

 

[timetraveller123]

okay thanks about that what about the second part
what does the question mean when it says the cylinder expands or contracts
does the area of the hollow part of the cylinder change or is it the b-a is changing ?

 

[rude man]

okay thanks about that what about the second part
what does the question mean when it says the cylinder expands or contracts
does the area of the hollow part of the cylinder change or is it the b-a is changing ?

Think about what's happening. Consider the cylinder comprising a large number of thin annular concentric cylindrical sections, each of differential volume 2πrL dr, a < r < b. So what forces act on a section close to r=b and what forces act on a section close to r = a?

Hint: what is the B field close to r=a and what is it close to r=b?

And what does that tell you about how the cylinder and/or the hollow area contract, assuming it's made of sufficiently pliable metal?

 

[timetraveller123]

no sir you have misunderstood the question i don't understand what the question is asking i would surely do what you say once you can make it clear to me what exactly the question is asking

 

[rude man]

no sir you have misunderstood the question i don't understand what the question is asking i would surely do what you say once you can make it clear to me what exactly the question is asking

You have to decide what's changing. What's changing is obviously associated with the forces acting on the part of the cylinder that's causing the change.
So yes, possibly, either a or b is changing, or both. You do see that if a is changing the hollow area is also changing, whereas if only b is changing then that area, which is πa², is not changing.

 

[timetraveller123]

sir try as i might i really don't see why would changing the area affect anything be it a or b that is being changed .
when a is changing only the area of the hollow is changing meaning the external magnetic flux through the cylinder is changing but so far the external magnetic field has not had any effect on the system
furthermore changing the b also only changed the solid cross sectional area but the question stated the current through it is constant 10MA so there might not be any change in the magnetic field caused the current
so my guess is that there would only be mechanical effects rather than any magnetic effects
for example maybe the pressure maybe taking effect due to the changes ??
i am sorry if i am still not seeing what you are driving at
would be of much help if could give some more hints

 

[rude man]

when a is changing only the area of the hollow is changing meaning the external magnetic flux through the cylinder is changing but so far the external magnetic field has not had any effect on the system

still doesn't

furthermore changing the b also only changed the solid cross sectional area but the question stated the current through it is constant 10MA so there might not be any change in the magnetic field caused the current

Mag. field is unchanged for r > b and r < a. But a or b or both might change ... up to you to figure out which.

so my guess is that there would only be mechanical effects rather than any magnetic effects
for example maybe the pressure maybe taking effect due to the changes ??
i am sorry if i am still not seeing what you are driving at
would be of much help if could give some more hints

I don't know how else to reply. Maybe you should read my previous posts a bit more thoroughly.

 

[timetraveller123]

this is the best i could try i am not lying
no matter which a or b changes the magnetic field at r>b is not changing due to the fact that the current through the wire is constant
the area of the wire just changes accordingly to which radius change that is simple enough
isn't that all the question is asking it asked how the magnetic field changes how the area changes and how the magnetic flux changes
i don't see the point in this question
maybe you should just give me the answer to just this part if i am still getting it wrong ?

 

[rude man]

this is the best i could try i am not lying
no matter which a or b changes the magnetic field at r>b is not changing due to the fact that the current through the wire is constant
the area of the wire just changes accordingly to which radius change that is simple enough
isn't that all the question is asking it asked how the magnetic field changes how the area changes and how the magnetic flux changes
i don't see the point in this question
maybe you should just give me the answer to just this part if i am still getting it wrong ?

1. for a < r < b what is the B field?
2. then what is the force on a thin annulus at r?
3. what might that force do to that annulus if the metal were soft enough to deform under pressure?
4. considering that the cylinder is made up of many such thin annuli, what is the cumuative effect on the cylinder?

 

[timetraveller123]

firstly sorry for the extremely later reply my o levels are coming and my prelimaries were going on so very sorry

ok i am answering your guiding questions

1)
$B = \frac{\mu_0 I_0}{2 \pi r} \frac{r^2 - a^2}{b^2 - a^2}\\ 2)\\ f = dI (L X B )\\$
3) the annulus would contract under the pressure
4) the cylinder would too contract under the pressure
is it correct?
now i know what my question is
the previously calculated 32 gpa of pressure where exactly is it acting on the outer curved surface area
how to calculate the effect of pressure on the cylinder?
and the question asking what is r_equilibrium seems to suggest to me that at the r equilibrium there is pressure from inside the cylinder acting outwards that is balancing the 32 gPa is that true and what force is causing that?
thanks for your understanding and sorry once again