# Why we calculate B of solenoid with this way?

• theodoros.mihos
In summary, the conversation discusses the calculation of magnetic fields using the Biot-Savart law for circular currents and the integration method for determining the total magnetic field from multiple rings in a solenoid. The conversation also brings up the importance of accurately setting up the integral and carefully examining the physics involved.
theodoros.mihos
We see the magnetic field calculation by Biot-Savart law for a circular current I on some distance z on the vertical axis throw it centre as:
$$B(z) = \frac{\mu_0I}{2}\frac{r^2}{(r^2+z^2)^{3/2}}$$
where r is the radius of circle.

When we use summation of N rings lays on L length, like solenoid, we integrate as:
$$B(z) = \frac{\mu_0nIr^2}{2}\int_{-L/2}^{L/2}\frac{dz}{[r^2+(z-z_0)^2]^{3/2}}$$

I think that ##B_i## for one ring gives to total ##B## an amount:
$$B_i = B(z_i) = \frac{\mu_0I}{2}\frac{r^2}{[r^2+(z_i-z_0)^2]^{3/2}}$$

How we take the integral ##\int\,dz## ?

Last edited:
How would you normally take the integral?
(You may want to try a trig substitution.)

Have you set up the integral correctly?
When you take the definite integral over a variable, it goes away - so you cannot have a function like ##f(z)=\int_a^b g(z)\;dz## because that implies that f is a function of z.

Usually you want to say something like - the ring between z=z' and z=z'+dz' contributes dB=something to the overall field at position z on the axis... then you put ##B=\int dB##. Then ##z'_i## is the position of the ith ring and you need to add them up.

##z_0## appears to be playing the role of ##z'## in your integral - so you need to integrate over that.

Have you examined the physics carefully ? - usually a solenoid has a finite number of rings in length L.
The integral implies that the rings are very thin.

Last edited:
theodoros.mihos
Simon Bridge said:
Usually you want to say something like - the ring between z=z' and z=z'+dz' contributes dB=something to the overall field at position z on the axis... then you put B=∫dBB=\int dB. Then ziz'_i is the position of the ith ring and you need to add them up.

I think this is the point I need. Thanks.

No worries.

## 1. Why do we use the formula B=μ0ni for calculating the magnetic field of a solenoid?

The formula B=μ0ni is derived from the Biot-Savart law, which is a fundamental equation in electromagnetism that describes the magnetic field produced by a current-carrying wire. The variables in the formula represent the permeability of free space (μ0), the number of turns per unit length of the solenoid (n), and the current flowing through the solenoid (i). This formula allows us to calculate the magnetic field at any point inside or outside the solenoid.

## 2. How does the number of turns in a solenoid affect the magnetic field strength?

The number of turns in a solenoid, represented by the variable n, is directly proportional to the magnetic field strength. This means that as the number of turns increases, the magnetic field strength also increases. This is because each turn of the solenoid contributes to the overall magnetic field, creating a stronger and more concentrated field.

## 3. What does the permeability of free space (μ0) represent in the formula for calculating the magnetic field of a solenoid?

The permeability of free space (μ0) is a physical constant that represents the ability of a vacuum to support the formation of a magnetic field. It is a measure of how easily a magnetic field can pass through a given material. In the formula B=μ0ni, μ0 is necessary to convert between units of current and magnetic field strength.

## 4. Can the formula for calculating the magnetic field of a solenoid be applied to other shapes or objects?

The formula B=μ0ni is specific to solenoids, which are long cylindrical coils with a uniform current flow. It can be used to calculate the magnetic field at any point inside or outside the solenoid, but it cannot be applied to other shapes or objects. Different formulas and methods must be used for different shapes and configurations.

## 5. Why do we use the Biot-Savart law to calculate the magnetic field of a solenoid instead of other methods?

The Biot-Savart law is a fundamental equation in electromagnetism and is specifically designed to calculate the magnetic field produced by a current-carrying wire. It takes into account the shape and size of the wire, as well as the distance from the wire, making it a more accurate and reliable method for calculating the magnetic field of a solenoid compared to other methods. Additionally, the Biot-Savart law can be applied to more complex configurations, making it a versatile tool for scientists and engineers.

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