Magnetic field inside a Solenoid

[Prashasti]

Homework Statement

Can I derive an equation for magnetic field inside a solenoid using the formula for magnetic field on the axis of a current carrying coil?

Homework Equations

B = μI/2r ( Magnetic field at the centre of a current carrying coil)

The Attempt at a Solution

B = μI/2r
Let the number of turns per unit length of the solenoid be 'n' and its length be 'a'
So,
B = μnaI/2r
Which is definitely not equal to μnI (Calculated using Ampere's Circuital Law)
What's wrong?[/B]

 

[ehild]

Homework Statement

Can I derive an equation for magnetic field inside a solenoid using the formula for magnetic field on the axis of a current carrying coil?

Homework Equations

B = μI/2r ( Magnetic field at the centre of a current carrying coil)

The formula refers to the magnetic field of an infinitely long straight current-carrying wire, at distance r from the wire. It is not valid in the centre of a loop.

 

[rude man]

Plus, the formula for the mag field around an infinite, current-carrying wire is B = μI/2πr, not what you wrote.

 

[Prashasti]

Well, my teacher has been using that formula...

The application of the Biot Savart Law on the centerline of a current loop involves integrating the z-component.

The symmetry is such that all the terms in this element are constant except the distance element dL , which when integrated just gives the circumference of the circle. The magnetic field is then

So, if we apply the same,
z = 0 (At the centre of the loop)
So, B = μI/2R
Isn't it correct??

 

[ehild]

It is correct for the magnetic field at the centre of a single loop. If you have a coil, all loops have their magnetic field inside the other loops. You should use the formula for B(z) and sum (integrate) the contributions of all loops.

 

[rude man]

Isn't it correct??

Yes, but tha's for a single loop, using Biot-Savart. Not the thing for here..

Rather than integrate per post #5 my hint is to form a closed loop going thru the entire solenoid middle and closing outside the solenoid. You can now apply Ampere's law to get B.
Hint: contributions to the integral outside the loop may be ignored.