Physics Help Forum use ampere-circuital law to establish the expression for magnetic field

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 Jun 8th 2010, 12:23 PM #1 Junior Member   Join Date: Dec 2008 Posts: 16 use ampere-circuital law to establish the expression for magnetic field to establish the expression for magnetic field at the centre of a current carrying loop using ampere-circuital law.. how do you do that? Using biot-sarvart law, we get B=mu*i/2*r...
Jun 26th 2010, 09:39 AM   #2
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 Originally Posted by ice_syncer to establish the expression for magnetic field at the centre of a current carrying loop using ampere-circuital law.. how do you do that? Using biot-sarvart law, we get B=mu*i/2*r...
That's pretty close, you are only missing a pi in the denominator. Amperes circuital law with Maxwell's correction states that the magnetic flux passing through some enclosed surface is directly related to a steady state current and a time changing electric flux passing through a surface, that need not be enclosed.

Since you have a current carrying loop I am assuming it only has a DC current flowing through it, if not then the following answer is not correct as I am leaving out the E flux that may be changing with time.

So,

INT{Bdl}s = muIs + mueo(dIe/dt) where mu is the permeability of free space, eo is the permittivity of free space, dIe/dt is the time rate of change of the electric flux through some surface, and INT{Bdl}s is the line integral enclosing some surface the B field is passing through. Now we are going to assume this is not a time changing current flowing through the loop, this allows us to ignore the rate of change in electric flux passing through some surface. So it reduces down to:

B*2*pi*r = muIs where r is the radius from the center of the loop to the loop itself. Therefore the magnetic flux passing through the center of the loop due to a constant current flow, Is, is a function of the area enclosed by that loop. So:

B = mu*I/(2*pi*r)

That should be the magnetic flux passing through the center of the loop due to a constant current flowing through the loop with intensity, I. If you are suppose to find the B field due to both a constant current flow and from a time changing electric field, do tell and I'll redo my answer to include that solution also.

Many Smiles,
Craig

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