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Why is it that integrating

**B ds**gives**BL**where**L**is the length of the solenoid?-
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- Thread starter kasse
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- #1

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

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alphysicist

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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:

[tex]

B L = \mu_0 N I

[/tex]

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:

[tex]

B = \mu_0 n I

[/tex]

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:

[tex]

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

[/tex]

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.

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