Magnetic field outside a solenoid

[Dell]

given a long solenoid, with a radius of R (R<<L), a current of I flows throught it. find the magnetic field at point A which is located at a distance of R from the solenoid on the solenoids line of symetry.

HOW DO I DO THIS??

the only thing i see that i think i need to do here is somehow use either biot savar or more likely ampere and solve it, but how?
the only problems i have had with solenoid were to find the field inside of it in which case i took a closed loop integral and used amperes law to get B=μNI/L=μnI

 

[jbsteele]

inside, but now i am stuck on how to find the magnetic field at a certain point outside the solenoid.... help please?

 

[nlightner]

where n=N/L and L is the length of the solenoid.

To find the magnetic field at point A, we can use Ampere's law again. This law states that the line integral of the magnetic field around a closed loop is equal to the permeability of free space (μ0) times the current enclosed by the loop. In this case, the loop we will use is a circle with a radius of R, centered at point A and perpendicular to the solenoid.

Since the radius of the solenoid (R) is much smaller than its length (L), we can assume that the magnetic field is uniform inside the solenoid and zero outside of it. Therefore, the only current enclosed by our loop is the current flowing through the solenoid, which is I.

Using Ampere's law, we can write:

∮B·dl = μ0I

Where ∮B·dl represents the line integral of the magnetic field around the loop, and μ0 is the permeability of free space.

Since the magnetic field is constant and parallel to the loop, we can simplify this equation to:

B∮dl = μ0I

The left side of the equation can be rewritten as the circumference of the circle times the magnetic field at point A, giving us:

2πRB = μ0I

Solving for the magnetic field, we get:

B = μ0I/2πR

Therefore, the magnetic field at point A is:

B = μ0I/2πR

This result is consistent with our previous understanding of the magnetic field inside a solenoid, where B = μ0nI, with n = N/L being the number of turns per unit length. In this case, since we are only considering one turn of the solenoid, the number of turns is 1 and the length is equal to the circumference of the loop, 2πR.

In summary, to find the magnetic field at point A outside a long solenoid, we can use Ampere's law and consider a circular loop centered at point A. This will give us the same result as using the equation for the magnetic field inside a solenoid, as long as the radius of the solenoid is much smaller than its length.