Solving Magnetic Field Inside Toroidal Solenoid of 0.50m Radius

[swankymotor16]

A toroidal solenoid has a central radius of 0.50 m and a cross-sectional diameter of 10 cm. When a current passes through the coil of the solenoid, the magnetic field inside the solenoid at its CENTER has a magnitude of 2.0 micro T. What is the largest value of the magnetic field inside the solenoid when this current is flowing?



cross-sectional radius= (0.10m)/2= 0.05m



I= (B)(2pi)(radius)/(permeability of free space 'mu subzero') = (2x10^-6 T)(2pi)(0.5m)/(4pi x10^-7)= 5.0 A



B=(permeability of free space)(current)/(2pi)(radius) = (4pi x10^-7)(5.0 A)/(2pi)(0.05m) = 5x10^-8 T



IS THIS CORRECT??



answers are:

1.8 micro T
0.50 micro T
2.8 micro T
2.2 micro T
3.5 micro T

 

[Simon Bridge]

A toroidal solenoid has a central radius of 0.50 m and a cross-sectional diameter of 10 cm. When a current passes through the coil of the solenoid, the magnetic field inside the solenoid at its CENTER has a magnitude of 2.0 micro T. What is the largest value of the magnetic field inside the solenoid when this current is flowing?

The question appears to be asking in the second part for information given in the first ... but I notice you have interpreted the two parts to refer to different radii...

cross-sectional radius= (0.10m)/2= 0.05m

Note: $2\pi r = \pi d$ ... just sayin.

I= (B)(2pi)(radius)/(permeability of free space 'mu subzero') = (2x10^-6 T)(2pi)(0.5m)/(4pi x10^-7)= 5.0 A

That would be:
$$I=\frac{2\pi r B_1}{\mu_0}$$
... finding the current due to the measured field in the center of the solenoid part? ... how does the turn-density figure into this?
I note that $\mu_0 I/d$ is the field at the center of a single current loop. You seem to have computed the field at the center of a solenoid length $2\pi r$ with one turn on it.

B=(permeability of free space)(current)/(2pi)(radius) = (4pi x10^-7)(5.0 A)/(2pi)(0.05m) = 5x10^-8 T

That would be:
$$B_2 = \frac{\mu_0 I}{2\pi R}$$
... Here I is from the first equation. I guess I have the same question: doesn't the field strength also depend on how many turns there are?

Notice that, from these equations:
$$\frac{B_2}{B_1} = \frac{r}{R}$$... you needn't have bothered with the extra calculation.

I think to be able to tell if you have the right answer or not you need to be clear about what you are calculating. Show us your reasoning.

 

[swankymotor16]

I can't understand the computer language in the final equations you are giving me. I calculated the magnetic field with the cross-sectional diameter using the current calculated in the first part of the problem. The cylinder must be one loop, since there is no specific information about it... But, I think I'm wrong, since the question wants a HIGHER magnetic field, and the answer gives me 0.005 nT :S

 

[Simon Bridge]

I can't understand the computer language in the final equations you are giving me.

Oh you mean when you quoted them?
I used tex markup so the equations should display (above) as clear graphics, but if you hit the "quote" button you'll get a bunch of code.

"\pi" renders as $\pi$ for example.

(If you see "itex" in square brackets instead then you have a problem.)

Have you copied out the question exactly as it is written?
I don't think it is clear - for instance, the largest value of the field inside the toroid for a given current is around the center right? But 2.0$\mu$T is not one of the correct answers. (Would also make it a trick question.) So I think something is missing ... maybe a context?

 

[swankymotor16]

I'm sorry, everything is there.

 

[TSny]

Have you studied Ampere's law? You can use it to find an expression for the field inside the toroid.

If you haven't covered Ampere's law, then hopefully the formula for the field of a toroid has been given to you in your textbook or in class.

From the formula you will be able to see where the field has a maximum value. Simon Bridge has given a good hint that you should consider the ratio of the maximum field to the field at the center of the cross section. In the ratio, things that you aren't given, like the number of turns on the toroid and the current, will cancel out.

 

[Simon Bridge]

I'm sorry, everything is there.

In that case - you have missed something in your reasoning. I think there is a confusion betwen the different places that could be called the "center of the toroid". It doesn't help that most diagrams show the loop for Amphere's Law drawn in pretty much the same places - making it look like these are the only places where the loop can be drawn.

You have a formula for the relationship between the field strength and a radius r.
Try using this to answer the following questions.

What is r the radius from?
At what values of r is B zero?
At what values of r is B non-zero?
What value of r corresponds to the center of the toroid?
Is B a maximum in the center?