Verify the Ampere's law for magnetic field of a point dipole moment `vecM=Mhatk`. Take C as the closed curve running clockwise along (i) z-axis from `z=agt0` to `z=R`, (ii) along the quarter circle of radius R and centre at the origin, in the first quadrant of x-z plane, (iii) along the x-axis from `x=R` to `x=a`, and (iv) along the quarter circle of radius a and centre at the origin in the first quadrant of x-y plane.

Explanation

From P to Q, every point on the z-axis lies at the axial line of magnetic dipole of moment M magnetic field induction at a point direction z from the magnetic dipole of moment is
`[B]=(mu_(0))/(4 pi)(2[M])/( z^(3))=(mu_(0)M)/(2piz^(3))`
(i) Along z-axis from P to Q.
`int_(P)^(Q)B.di=int_(P)^(Q)B,dicos0^(@)=int_(a)^(R)Bdz`
`=int_(a)^(R)(mu_(0))/(2pi)(M)/( z^(3))dz=(mu_(0)M)/(2pi)((-1)/(2))((1)/(R^(2)-(1)/( a^(2)))`
`=(mu_(0)M)/(4 pi)((1)/(a^(2))-(1)/(B^(2)))`
(ii) Along the quarter circle QS of radius R as given in the figure below

The point A lies on the equational line of the magnetic dipole of moment `M sin theta` Magnetic field at point A on the circular are is
`B=( mu_(0))/(4pi)(Msin theta)/(R^(3)),di=Rdtheta`
`therefore intB.di=intB,di cos theta=int_(Q)^((pi)/(2))(mu_(0))/(4pi)(Msin theta)/(R^(3))Rdtheta`
Circular arc `=(mu_(0))/(4pi)(M)/(R)(-cos theta)_( O)^(pi//2)=(mu_(0))/(4pi)(M)/(R^(2))`
(iii) Along x-axis over the path ST, consider the figure given ahead

From figure every point lies on the equational line of magnetic dipole Magnetic field induction at a point distance x from the dipole is
`B=(mu_(0))/(4pi)(M)/(x^(3))`
`therefore int_(S)^(T)Bdi=int_(n)^(a)-(mu_(0)M)/(4pix^(3))d=0[because "angle between"(-M)"and" di "is" 90^(@)]`
(iv) Along the quarter circle TP of radius a consider the figure given below

From case (ii), we get the line integral of B along the quarter circle TP of radius a is
Circular arc TP `intB.di=int_(pi//2)^(0)(mu_(0))/(4pi)(Msin theta)/(a^(3))adtheta`
`=(mu_(0))/( 4pi)(M)/(a^(2))[-costheta)_(pi//2)^(0)`
`=(-mu_(0))/(4pi)(M)/(a^(2))`
`therefore ointB,di=int_(P)^(Q)B.di+int_(Q)^(S) B.di+int_(S)^(T)B.di+int_(T)^(P)B.di`
`=(mu_(0)M)/(4)[(1)/(a^(2))-(1)/(R^(2))]+(mu_(0))/(4pi)(M)/(R^(2))+0+(-(mu_(0))/(4pi)(M)/(a^(2)))=0`