HELP Electromagnetism Problems

[_gaussian]

Hey everyone,

I just wanted to double check some solutions and inquire about a problem.

Homework Statement

1. A toroid is a coil of wire wrapped around a doughnut-shaped ring (a torus). For a tightly wrapped toroid with $$N$$ turns, the magnetic field lines of $$\vec{B}$$ form concentric circles inside the toroid, and the field is zero outside. Use Amperes law to find an expression for the magnetic field strength $$\vec{B}$$ at a radial position r from the axis of the toroid.
2. A 100 cm diameter cyclotron uses a 1000 volt oscillating potential difference between the “dees” to accelerate protons.
   (a) What is the maximum kinetic energy of a proton in the beam emerging from the cyclotron
   if the magnetic field strength is 0.6 Tesla?
   (b) Estimate how many revolutions the proton makes before it emerges from the cyclotron.
3. Two long solenoids are nested on the same axis as shown in the figure below. They carry
   identical currents, but in opposite directions. If there is no magnetic field inside the inner
   solenoid, what can you say about the numbers of turns per unit length, $$n_{outer}$$ and $$n_{inner}$$, of the two solenoids? Which one, if either, has the larger value?
4. A messy loop of limp, flexible wire is placed on a frictionless tabletop and anchored at points a and b, as shown in the figure below. If a large current I is now passed through the wire, how will the wire move? What shape do you think the wire will assume, and why? Explain your answer with words and diagrams.

Homework Equations

For an toroid:
$$\vec{B} = \frac{\mu_0 iN}{2\pi r}$$
For an ideal solenoid:
$$\vec{B} = \mu_0 in$$
(where n is the number of turns per unit length)
For a circulating charged particle:
$$r = \frac{mv}{|q|B}$$
For a cyclotron:
In resonance
$$f = f_{osc}$$
$$|q|B = 2\pi mf_{osc}$$

The Attempt at a Solution

1.
Using the Ampere's Law:
$$\oint \vec{B}\cdot ds = \oint \vec{B}\cos{\theta}\cdot ds = \mu_0 i_{enc}$$
Since:
$$i_{enc} = iN$$
and,
$$\oint \vec{B}\cdot ds = \vec{B}(2\pi r)$$
Then,
$$\vec{B}(r) = \frac{\mu_0 iN}{2\pi r}$$

2.
I've tried to use the following equations but end up getting different answers for $$\vec{v}$$

$$\frac{1}{2}m\vec{v}^2 - qV = 0$$
As well as rearranging:
$$r = \frac{mv}{|q|B}$$

What to do?

3.
Since $$R_{out} > R_{in}$$ and $$\vec{B}_{out}= \vec{B}_in$$
Equating them (using the equation with dependence on R)
$$n = \frac{N}{l}$$
[tex]nl = N[tex]
$$\frac{\mu_0 in_{out}l}{2R} = \frac{\mu_0 in_{in} }{2R}$$
Proportionality shows:
If $$R_{out}> R_{in}$$ then for $$\vec{B}_{out} = \vec{B}_{in}$$ then $$n_{in}>n_{out}$$.

4.
Intuitively, I think the wire will remain still, but I think I must calculate something with the line integral?

Thank you for your help!

 

[_gaussian]

Please help!

 

[_gaussian]

Please help!