How Does Integrating Magnetic Field in a Solenoid Result in BL?

[kasse]

Why is it that integrating B ds gives BL where L is the length of the solenoid?

 

[alphysicist]

The usual method of using Ampere's law does not directly give the length of the solenoid, because the usual assumption is that the solenoid is infinitely long.

The Amperian loop is usually a rectangle, with a side of length L inside and outside the solenoid, and there are N turns of the solenoid passing through the loop. Then Ampere's law gives:

$$B L = \mu_0 N I$$

But the specific values of N and L were rather arbitrary in that they depended on how big the loop is; if the loop's side inside the solenoid were doubled, both N and L would double. To get something useful, they combine this in terms of n, the turns per length:

$$B = \mu_0 n I$$

Now when you use this equation for a real solenoid, if they give you the total turns and total length of the solenoid, you can go back and plug these in:

$$B = \mu_0 \frac{N}{L} I$$

but these values of N and L (total turns and total length) are technically not the N and L (turns going through Amperian loop and length of one of the sides of the Amperian loop) that you use in Ampere's law.

Did that answer what you were asking?

 

[Moetasim]

Ampere's law states that the magnetic field around a closed loop is proportional to the electric current passing through the loop. In the case of a solenoid, which is a tightly wound coil of wire, the current passing through the loop is equal to the current passing through each individual turn of the coil. This results in a strong and uniform magnetic field inside the solenoid.

When we integrate the magnetic field (B) along the length (L) of the solenoid, we are essentially summing up the contributions of the magnetic field from each individual turn of the coil. This is because the magnetic field from each turn is parallel to the length of the solenoid and thus adds up in a linear fashion.

Since the magnetic field inside the solenoid is constant, this integration can be simplified to B multiplied by the length of the solenoid (BL). This is why integrating B ds gives BL, where L is the length of the solenoid.

In summary, integrating B ds along the length of a solenoid gives BL because the magnetic field inside the solenoid is constant and the contributions from each turn of the coil add up linearly. This is a fundamental concept in understanding the behavior of solenoids and is essential in many applications of electromagnetism.