Question

using ampere circuital low find the magnetic flux density at the centre of long solenoid carrying current.​

Answer 1

Answer:

Consider a long straight conductor carrying current I perpendicular to the page in upward direction as shown below in the figure

Proof Of Ampere's Law  

From Biot Savart law, the magnetic field at any point P which is at a distance R from the conductor is given by

 

Direction of magnetic Field at point P is along the tangent to the circle of radius R withTh conductor at the center of the circle

For every point on the circle magnetic field has same magnitude as given by  

 

And field is tangent to the circle at each point

The line integral of B around the circle is

 

since ∫dl=2πR ie, circumference of the circle so,

 

This is the same result as stated by Ampere law

This ampere's law is true for any assembly of currents and for any closed curve though we have proved the result using a circular Amperean loop

If the wire lies outside the amperion loop, the line integral of the field of that wire will be zero  

 

but does not necessarily mean that B=0 everywhere along the path ,but only that no current is linked by the path

while choosing the path for integration ,we must keep in mind that point at which field is to be determined must lie on the path and the path must have enough symmetry so that the integral can be evaluated

Explanation: