Deciding the direction of magnetic field of a solenoid

[Pushoam]

Homework Statement

Decide the direction of Magnetic field of a very long solenoid consisting of n turns per unit length on a cylinder with current I flowing through.

Homework Equations

The Attempt at a Solution

Deciding direction of the magnetic field:
Considering cylindrical coordinates with taking axis of cylinder as z - axis,
The direction of magnetic field at any point has to be perpendicular to $\hat \phi$ as the current is moving in this direction and Biot -Sawrt Law tells us that magnetic field due to a steady current is perpendicular to the direction of current. So, the magnetic field can have a component in $\hat s$ or $\hat z$ direction.
If the magnetic field has a component in $\hat s$ direction, then this means that the magnetic field is originating from somewhere or terminating at somewhere i.e. existence of magnetic monopoles i.e.divergence of the magnetic field then would be non - zero.

This can be shown by taking a very small length cylindrical Gaussiasn surface concentric to the axis of the solenoid. The length is so small that the variance of magnetic field over z could be ignored. Then it could be shown that the total flux through this surface is not 0. HEnce the component of magnetic field in the $\hat s$ direction is 0.
Thus, magnetic field has to be along $± \hat z$ direction depending on the direction of current.

 

[berkeman]

Seems like you could just use the Righthand Rule to answer the question. Is it meant to be more complicated than that?

http://www.aplusphysics.com/courses/honors/magnets/images/2ndRHR.png