Exploring Magnetic Fields in a Coiled Wire

[Ryoblck]

Homework Statement

A 1.0 m piece of wire is coiled into 200 loops and attached to a voltage source as shown.

A. Find the strength of the magnetic field inside the coil if V = 100 V and R = 40 Ω.
B. Which direction does the magnetic field point?
C. The wire is then uncoiled and re-wrapped so that the cross-sectional area of the coil is twice what it was previously, though the length stays the same. What is the new magnetic field strength inside the coil?
D. The entire coil of wire is now placed into an external magnetic field as shown below. Which direction will the coil first begin to rotate?

Homework Equations

B=(u0*N*I)/L
u0=4pi*10^-7 TM/A

The Attempt at a Solution

I solved the first two parts of the problem.
B=(4pi*10^-7 TM/A*200*2.5A)/(0.1m)= 6.28e-3 T and the magnetic field points down with the right hand rule. Now part C has some weird wording to the problem that is throwing me off. Also would not understand part D really. Don't remember ever going over it before.

 

[Sentin3l]

For part C, they are saying that they unwrapped the coil and then re-wound it into a second coil with twice the cross-sectional area (i.e. the second coil has a bigger radius). Consider how that would effect how many loops are in the coil.

For D, I would suggest drawing what the magnetic field around a coil would look like (your analysis in part B may be incorrect), having done that, see if you can work out how the coil's magnetic field interacts with the applied magnetic field.

 

[Ryoblck]

So the coil in part C would result in only 100 loops due to the bigger radius? And for part B, I think I was using the wrong angle. So possibly the magnetic field points upwards?

 

[rude man]

If the cross-sectional area is doubled, what happens to the circumference? What is the relation between number of turns, wire length and circumference and so how many turns do you wind up with?

The magnetic field inside the coil does not point upward or downward. Use Ampere's law to find the direction. Then think of the coil as a permanent magnet with defined N and S poles and also think of the externally applied field as being generated by a large magnet with defined N and S poles. Then all you need to know, which I know you do, that a N is attracted to a S and repelled by another N, and vice-versa.

 

[Ryoblck]

Ok I understand part B and D now. Regarding the cross-sectional area, would the result of the larger circumference equal to 100 turns exact? Because since we doubled the cross-sectional area, the number of turns have to be divided by two. Or am I looking in the wrong direction? I don't think I quite get the relation between everything.

 

[Sentin3l]

Doubling the cross-sectional area does not lead to doubling the radius, halving the loops, etc. Try to work this out.

Well your know first that circumference and area are given by the following (assuming a perfect circle):

[itex] C = 2 \pi r [/itex]

[itex] A = \pi r^{2} [/itex]​

I would suggest trying to find a way to express circumference in terms of area. Once your have that, find the new circumference of your coil. Then it should be smooth sailing to find how may loops are in the coil (think of how much wire is used per loop).

 

[Ryoblck]

But there is no radius given. So how will I find it? No area given either but the length of the uncoiled wire and coiled wire.

 

[rude man]

But there is no radius given. So how will I find it? No area given either but the length of the uncoiled wire and coiled wire.

The original radius you can deduce by the length of the wire and the number of turns.

The new radius is also deducible as it's related by the ratio of cross-sectional areas.

 

[Ryoblck]

What formula do I use? I don't know any formula that finds the radius with the length and the number of turns given.

 

[rude man]

I give you a 1 m long wire, you wrap it all around a (right circular) cylindrical object, you count the number of turns = 200, and you can't figure out the circumference of the cylinder?

 

[Ryoblck]

The formula I know to find the circumference of the cylinder involves the radius. 2(pi)r is the circumference... So what am I missing?

 

[rude man]

Take the 1 m wire, wrap it around a radius of R once, how much wire did that take?

 

[Ryoblck]

1/(piR^2) is the answer. Meaning I subsititute that into the circumference formula?

 

[rude man]

1/(piR^2) is the answer. Meaning I subsititute that into the circumference formula?

Your answer should have dimensions of length, not 1/area,don't you think?
Again: how much wire did it take to wrap 1 turn around the cylinder?

 

[Ryoblck]

The length of wrapped wire is 100cm. With 200 turns, we can determine that is takes .5cm for each turn to wrap the wire. How would I include the length of the wire into the equation? I'm not seeing this at all.

 

[rude man]

OK, it takes 0.5cm to wrap one turn. Does that sound like it equals the circumference of the coil?

 

[Ryoblck]

Oh my gosh. I feel so dumb right now. With the given information, the original radius is .080cm. With this, the cross-sectional area is doubled. The cross-sectional area was .0201cm and now is .0402cm. Now that I have the new area and the same length, I found that 100cm/.0402cm would equal to the new amount of turns. The answer is given as 2488 turns which cannot be right. How do I calculate the number of turns? I have an equation in the physics textbook but it requires given information of B solenoid.

Edit: Just realized I miscalculated the length of wire when changing units. 1m goes to 100cm and .1m is 10cm. That makes the circumference .05cm and the radius is .008cm. Just one decimal difference. The original area is .08cm and the cross-sectional is .158cm. With the length of the wrapped wire, I divided it with a new cross-sectional finding that it takes 63 turns. But using this, I used the original area I got 125 turns. So is this partially correct?

 

[rude man]

Oh my gosh. I feel so dumb right now. With the given information, the original radius is .080cm. With this, the cross-sectional area is doubled. The cross-sectional area was .0201cm and now is .0402cm. Now that I have the new area and the same length, I found that 100cm/.0402cm would equal to the new amount of turns. The answer is given as 2488 turns which cannot be right. How do I calculate the number of turns? I have an equation in the physics textbook but it requires given information of B solenoid.

Area is not expressed in cm. Your number for the old & new areas are OK.

N = 100cm/circumference, not 1/area. Recompute the new N.

 

[Ryoblck]

Area is not expressed in cm. Your number for the old & new areas are OK.

N = 100cm/circumference, not 1/area. Recompute the new N.

Wait but when I converted 0.1m to cm it actually was 10cm...

Edit: Agh I'm getting confused in my own math. I'm lost but I'll just try and follow what you're doing. With 100/circumference, the equation would be 100/1 making it 100 turns. That sounds correct to me now.