# Derivation of magnetic field of a Solenoid: Biot savart law

• Conductivity
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.
Conductivity
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?

Page 8, Example 9.

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.

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.

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.

## 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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