Calculating Magnetic Fields in Solenoids vs. Current Loops and Coils

[The Head]

Homework Statement

There is a solenoid, with field B at its center, radius r, length L and maximum current I. Calculate the minimum number of turns the solenoid must have.

Homework Equations

Ampere's Law: (closed)∫B$\cdot$dl=$\mu$I
n=N/L
Biot-Savart: B=$\mu$IN/(2$\pi$r

The Attempt at a Solution

Solving Ampere's Law gives you BL=$\mu$IN and so N=BL/($\mu$I).

My question is why can't this same solution be found by solving with Biot-Savart, which is used for coils, according to my book? In that case, my answer is N=2Br/($\mu$I), and thus is off by a factor of 2r/L, or the diameter over the length of the solenoid.

Can someone please explain this to me? Thank you!

 

[anathema]

Hello,

Thank you for your question. The reason why the solution using Biot-Savart is different from the solution using Ampere's Law is because they are based on different principles.

Ampere's Law is based on the concept of a closed loop and relates the magnetic field around a closed loop to the current passing through the loop. This law is applicable to any closed loop, including a solenoid.

On the other hand, Biot-Savart law is based on the principle that the magnetic field at a point is directly proportional to the current at that point. This law is applicable to a single current carrying wire or a point charge, and is not directly applicable to a solenoid.

When using Biot-Savart law to find the magnetic field inside a solenoid, we must consider the contributions of each individual current element along the length of the solenoid. This is why the solution using Biot-Savart law is different from the solution using Ampere's Law.

I hope this explanation helps to clarify the difference between the two solutions. Please let me know if you have any further questions.