What Causes the Magnetic Field Outside a Realistic Solenoid?

[Irid]

Homework Statement

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.

Homework Equations

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?

 

[Gokul43201]

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,
$$I' = I \frac{d}{2\pi a} = \frac{I}{2\pi a N}$$

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?

 

[mgb_phys]

a realistic circular solenoid of infinite length

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

Sorry - not a very helpful comment, i know.

 

[Irid]

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?

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.

 

[chwie]

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