Magnetic Field of a Solenoid Using Ampere's Law
---- CLAIMED BY JAKE WEBB ----
This page explains how to use Ampere's Law to solve for the magnetic field of a solenoid.
The Main Idea
A solenoid is a long coil of wire with a very small diameter, often used to make electromagnets due to their ability to create strong magnetic fields. The magnetic field can be easily calculated along the axis of the solenoid using Ampere's Law, and the magnitude and direction of the field is constant throughout the entirety of the solenoid, excluding the ends.
As you can see in this model, at the ends of a solenoid the magnetic field begins to point outward at angles from the axis, with some of the field pointing directly perpendicular. In he middle, however, the field is constant throughout the entirety of the interior and is parallel to the axis.
A Mathematical Model
If there are [math]\displaystyle{ N }[/math] loops of wire the compose a solenoid of length [math]\displaystyle{ L }[/math], and we know that Ampere's Law for Magnetism gives us the form:
[math]\displaystyle{ {\oint\,\vec{B}•d\vec{l} = μ_{0}∑I_{inside path}} }[/math]
As stated above, the magnetic field inside a solenoid is constant and parallel to [math]\displaystyle{ dl⃗ }[/math], therefore Ampere's Law can be simplified to:
[math]\displaystyle{ BL=μ0NI }[/math]
By simply solving for [math]\displaystyle{ B }[/math] we can find the equation for the magnetic field of a solenoid:
[math]\displaystyle{ {B = \frac{μ_{0}NI}{L}} }[/math]
The direction can be found using the Right Hand Rule with your fingers curling around in the direction of the current and your thumb pointing in the direction of the magnetic field. It will always be along the axis for a solenoid.
A Computational Model
Examples
Simple
Find the magnetic field produced by the solenoid if the number of loops is 400 and current passing through on it is 5 A.( Length of the solenoid is 40cm)
This example is very simple, as all of the variables of the equation are provided. All you need to do is plug them in and find the direction using the right hand rule.
Middling
A solenoid has a magnetic field of 5x10-4 T, 3000 turns, and is 30 cm long. What is the current through the solenoid?
This example is just a reverse solving of the equation derived above for the magnetic field of a solenoid. It is not munch more difficult than the previous example.
Difficult
See Also
Magnetic Field of a Long Thick Wire Using Ampere's Law
Magnetic Field of Coaxial Cable Using Ampere's Law
Magnetic Field of a Toroid Using Ampere's Law
Further Reading
Chabay, Sherwood. (2015). Matter and Interactions (4th ed., Vol. 2). Raleigh, North Carolina: Wiley.
External Links
References
Chabay, Sherwood. (2015). Matter and Interactions (4th ed., Vol. 2). Raleigh, North Carolina: Wiley. Pg 887-888.