What is the Magnetic Field of a Long Solenoid?

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SUMMARY

The magnetic field of a long solenoid can be calculated using Ampere's Law, with the formula Bz = μ₀NI, where Bz is the magnetic field, μ₀ is the permeability of free space, N is the number of turns per unit length, and I is the current flowing through the solenoid. At the center of the solenoid, the magnetic field is uniform and equal to μ₀NI. For the ends of the solenoid, symmetry can be utilized to understand that the magnetic field is not uniform and decreases towards the ends, requiring a different approach to calculate accurately.

PREREQUISITES
  • Ampere's Law
  • Understanding of magnetic fields
  • Basic knowledge of solenoids
  • Concept of magnetic field symmetry
NEXT STEPS
  • Study the derivation of Ampere's Law in detail
  • Learn about the magnetic field of finite solenoids
  • Explore the concept of magnetic field lines and their behavior near solenoid ends
  • Investigate applications of solenoids in electromagnetic devices
USEFUL FOR

Students studying electromagnetism, physics educators, and anyone interested in understanding the behavior of magnetic fields in solenoids.

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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 Bz. 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 Bz=μoNI, so we can say that the magnetic field at the center is also μoNI. 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.
 
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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).
 
mfb said:
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?
 
The middle is just the sum of two ends.
 

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