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CBSE Class 12-science Answered

State ampere’s circuital law and express it mathmatically.

Asked by Topperlearning User | 28 Apr, 2015, 03:37: PM
Expert Answer

Ampere’s law states that the path integral or line integral over a closed loop of the magnetic field produced by a current distribution is given by

begin mathsize 11px style contour integral space straight B with rightwards arrow on top space times space dl with rightwards arrow on top space equals space straight mu subscript 0 space straight I end style                                                                                                                                where I refers to the current enclosed by the loop.

Ampere’s law is a useful relation that is analogous to Gauss’s law of electrostatics. It is a relation between the tangential component of magnetic field at points on a closed curve and the net current through the area bounded by the curve.

 

To evaluate the expression for begin mathsize 11px style contour integral space straight B with rightwards arrow on top space times space dl with rightwards arrow on top end style, let us consider a long, straight conductor carrying a current I, passing through the centre of a circle of radius r in a plane perpendicular to the conductor.

                            

According to Biot-Savart law of magnetism, the field has a magnitude begin mathsize 11px style fraction numerator straight mu subscript 0 straight I over denominator 2 πr end fraction end style at every point on the circle, and it is tangent to the circle at each point.

The line integral of begin mathsize 11px style straight B with rightwards arrow on top end style  around the circle is

 

             begin mathsize 11px style contour integral space straight B with rightwards arrow on top times dl with rightwards arrow on top space equals space contour integral space fraction numerator straight mu subscript 0 straight I over denominator 2 πr end fraction dl space equals space fraction numerator straight mu subscript 0 straight I over denominator 2 πr end fraction contour integral dl end style  

Since begin mathsize 11px style contour integral space dl with rightwards arrow on top space equals space 2 πr end style is the circumference of the circle,

Therefore, begin mathsize 11px style contour integral straight B with rightwards arrow on top times dl with rightwards arrow on top space equals space straight mu subscript 0 straight I end style                                                            

Answered by | 28 Apr, 2015, 05:37: PM
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