# Jackson 5.2b - realistic solenoid

1. Jul 26, 2007

### Irid

1. The problem statement, all variables and given/known data
There is a realistic circular solenoid of infinite length. Current I flows in it, and there are N turns per unit length. The problem asks to show that just outside the solenoid there exists a magnetic field identical in magnitude and direction as that of a single wire on the axis, carrying a current I.

2. Relevant equations

3. The attempt at a solution
To me it seems clear that the effect is caused by some current flowing in z direction, along the cylinder, that is. While azimuthally the current travels once around the circumference, upwards the current travels the distance equal to diameter of the wire, and from proportionality,
$$I' = I \frac{d}{2\pi a} = \frac{I}{2\pi a N}$$
That is the current flowing upwards. Using Ampere's law, the azimuthal magnetic field is
$$B_{\phi} = \frac{\mu_0 I'}{2\pi a} = \frac{\mu_0 I}{2\pi a} \frac{1}{2\pi aN}$$
Somehow, it doesn't resemble the answer suggested in the problem. What did I do wrong?

2. Jul 26, 2007

### Gokul43201

Staff Emeritus
This relationship is not justified. If you draw an Amperean loop that is circular and concentric with the axis of the solenoid, what current passes through the plane of this circle?

3. Jul 26, 2007

### mgb_phys

In what reality? Or is 'realistic' solenoid a code for some other property?

Sorry - not a very helpful comment, i know.

4. Jul 27, 2007

### Irid

Good point. Now I see my mistake. Ampere's law doesn't care in which direction the current flows, it only must cross the surface.
Thanks.

5. Nov 9, 2011

### chwie

In realistic solenoid means that is a helix, that mean that there is a component of the current in the z direction.