What is the Magnetic Field of a Long Solenoid?

[shinobi20]

Homework Statement

A long solenoid has current I flowing through it, also denote N as the turns per unit length. Take its axis to be the z-axis, by symmetry the only component of the magnetic field inside is B_z. Find the magnetic field at the center of the solenoid (on the axis). Also, find the magnetic field at the ends of the solenoid.

Homework Equations

Ampere's Law

The Attempt at a Solution

For the first part, since the solenoid is long we can approximate the magnetic field inside to be uniform and is given by B_z=μ_oNI, so we can say that the magnetic field at the center is also μ_oNI. I'm not sure if my argument is correct but based on my understanding, from the uniformity of the B field inside, it should be the same everywhere inside. Can anyone kindly tell me if this is correct? Any suggestions and insights?

For the second part I don't have any idea on how to start.

 

[mfb]

Right.

For the second part, you can use symmetry: imagine you are in the middle of the solenoid, and split it in two parts: "below" and "above" (along the z axis).

 

[shinobi20]

Right.

For the second part, you can use symmetry: imagine you are in the middle of the solenoid, and split it in two parts: "below" and "above" (along the z axis).

If you split the solenoid into two parts, then the total magnetic field inside each half solenoid would also be half as much but I can't see how I can tell the magnetic field at both ends by imagining to be in the middle. Can you expound more?

 

[mfb]

The middle is just the sum of two ends.