Derivation of magnetic field of a Solenoid: Biot savart law

In summary, the conversation discusses the use of Biot Savart's law for infinitely narrow wires and deriving the magnetic field of a solenoid. The substitution of ##k = n_o dx## is necessary, but may raise questions about the idealization of the solenoid and the use of infinitesimals. However, the derivation in the provided pdf is considered valid.
  • #1
Conductivity
87
3
Hello,

I have seen that biot savart's law works for infinitely narrow wires:
"The formulations given above work well when the current can be approximated as running through an infinitely-narrow wire."

When I wanted to derive the magnetic field of a solenoid, I had to do this substitution:
##n_o = N/L##

## k = n_o dx ##
Where k is the number of turns per dx.. But shouldn't K be an integer? so I can substitute it in the formula for circular coils. That means I have infinite number of turns and turn density of something like ## \frac{a}{dx} ## where a is an integer.

Is there is something wrong or that this is the idealization that we do to the solenoid? Wouldn't it be way off the correct value?

If you want the proof, http://nptel.ac.in/courses/122101002/downloads/lec-15.pdf
Page 8, Example 9.

Thank you in advance.
 
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  • #2
Conductivity said:
But shouldn't K be an integer?
No reason. If you have an 11 cm coil with 10 turns, you have 90.91 turns/m.

Note: never, never ever write something like "##
\frac{a}{dx}## with ##a## finite".
##dx## is universally seen as an infinitesimal (something that goes to zero). So ##
\frac{a}{dx}## does not exist.
 
  • #3
BvU said:
No reason. If you have an 11 cm coil with 10 turns, you have 90.91 turns/m.
Don't we idealize a solenoid as a number of circular coils?

https://i.imgur.com/RCO3qcQ.png

If we take a dx piece of this solenoid and treat at as k number of coils,

The equation for a single coil is:
## B = \frac{ u_o i R^2}{2 ( R^2 + x^2)^{\frac{3}{2}}} ##
If dx has 2 turns then we multiply by 2, If it has k turns then we multiply by k. But a dx piece can't have a 2.5 or a fraction of a coil (It doesn't make sense), Can it?
That is why I said ## k = n_o dx ## has to be an integer.
 
  • #4
Conductivity said:
Don't we idealize a solenoid as a number of circular coils?
We calculate the B field as if it were caused by a cylindrical sheet of current. That is a very good approximation (example 8 already indicates that).

Conductivity said:
If dx has 2 turns then we multiply by 2, If it has k turns then we multiply by k. But a dx piece can't have a 2.5 or a fraction of a coil
Again: do NOT use infinitesimals this way -- it will get you into trouble. A infinitesimal ##dz## piece has zero turns and nevertheless contributes to B with an infinitesimal contribution ##dB## proportional to ##{N\over L}##. It is only when you integrate ##dB## over a finite range in ##z## that you get a finite result. The derivation (it's not a proof) in the pdf is just fine.
 

Related to Derivation of magnetic field of a Solenoid: Biot savart law

1. What is the Biot-Savart law?

The Biot-Savart law is a fundamental principle in electromagnetism that describes the relationship between an electric current and the magnetic field it produces. It states that the magnetic field at a point in space is directly proportional to the current, the length of the current-carrying wire, and the sine of the angle between the wire and the point in space.

2. How is the Biot-Savart law applied to a solenoid?

In the case of a solenoid, the Biot-Savart law is used to calculate the magnetic field at a point along the axis of the solenoid. The law states that the magnetic field at a point is equal to the sum of the magnetic fields produced by each individual current element along the length of the solenoid.

3. What is the direction of the magnetic field inside a solenoid?

The magnetic field inside a solenoid is directed along the axis of the solenoid and is parallel to the direction of the current flowing through it. This means that the magnetic field lines inside a solenoid are uniform and parallel to each other.

4. Can the Biot-Savart law be used to calculate the magnetic field outside of a solenoid?

Yes, the Biot-Savart law can be used to calculate the magnetic field at any point in space, both inside and outside of a solenoid. However, the magnetic field outside of a solenoid may be weaker and more complex due to the interaction with external magnetic fields.

5. How does the number of turns in a solenoid affect the magnetic field?

The magnetic field inside a solenoid is directly proportional to the number of turns in the solenoid. This means that increasing the number of turns in a solenoid will also increase the strength of the magnetic field. However, the shape and size of the solenoid also play a role in determining the strength of the magnetic field.

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