Magnetic Field of a Solenoid
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A Solenoid is a type of electromagnet, which consists of a coil tightly wound into a helix. Usually it produces a uniform magnetic field when an electric current is run through it. The purpose of a solenoid is to create a controlled magnetic field.
The Main Idea
The purpose of this application is to explore another way to apply the Biot-Sarvart law. The magnetic field is uniform along the axis of the solenoid, when electric current is run through it. The solenoid has has to have coils much larger than the radius.
A Mathematical Model
This is the formula for the magnetic field inside a long solenoid: [math]\displaystyle{ B = {\mu _{0}} \frac{NI}{L} }[/math] This formula is used inside a long solenoid, when the radius of the solenoid is much smaller than the length. B is the magnetic field. [math]\displaystyle{ {\mu _{0}} }[/math] is a constant. N is the number of loops in the solenoid. L is the length of the solenoid.
To determine the direction of the magnetic field: use your right hand and curl your fingers towards the direction of the current and the direction that your thumb is pointing to is the direction of the magnetic field.
A Computational Model
Examples
Simple
A simple example of this
Difficult
To solve for the magnetic field of a solenoid, you can use a four step process.
Step 1: Cut up the distribution into pieces and Draw the [math]\displaystyle{ \Delta \vec{B} }[/math], Think about this: a solenoid the length of L that's made up for N circular loops tightly wound, each with a radius of R and a conventional current in the loops is I.
We want to find the magnetic field contributed by each of the loops at any location along the axis of the solenoid.
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Step 2: Write an Equation for the Magnetic Field Because of One Piece. The origin is located at the center of the solenoid. We find the integration variable [math]\displaystyle{ \Delta {x} }[/math], which is given by the location of a single piece. d-x is given by the distance from the loop to the observation location.
(picture)
We also have to consider the number of loops so there are N/L loops so the loops per [math]\displaystyle{ \Delta {x} }[/math] is (N/L)[math]\displaystyle{ \Delta {x} }[/math]. So from the magnetic field formula we have this so far:
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Step 3: Add Up the Contributions of All the pieces
Connectedness
Solenoids have many applications and they are a very common aspect of our daily lives. They can be found in electric locking mechanisms, medical equipment, cars, air conditioning systems, as well as many other devices. To learn more about the applications of Solenoids in detail, you can go to this internal link: Solenoid Applications
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