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  • Verify the Gauss’s law for magnetic field of a point dipole of dipole moment m at the origin for the surface which is a sphere of radius R.

Verify the Gauss’s law for magnetic field of a point dipole of dipole moment m at the origin for the surface which is a sphere of radius R.



 

Answers (1)

To prove Gauss'Law B.ds=0.
Dipole's Magnetic moment at origin O is along the z-axis.
Assume P to be a point at a distance r from Origin and OP is at an angle θ with the z-axis.
Component of M along OP=M\cos\theta 
Now, the magnetic field induction at P due to the dipole of moment M \cos\theta is B

=\frac{\mu _0}{4\pi} 2M\frac{\cos\theta}{r^3 }\widehat{r}
From the diagram, r is the radius of a sphere with centre at O lying in yz-plane.
An elementary area dS is taken at P   dS=r\left (r\sin \theta d\theta \right ) \widehat{r}=\left (r^2\sin\theta d\theta \right ) \widehat{r }

\oint B.ds=\oint \frac{\mu _{0}}{4 \pi }2M\frac{\cos \theta }{r^{3}}\widehat{r}(r^{2}\sin \theta d \theta )\widehat{r}
=\frac{ \frac{\mu _0}{4\pi}M}{r} \int_{0}^{2\pi }2\sin \theta \cos\theta d\theta

=\frac{\mu _{0}}{4\pi}\frac{ M}{r} \int_{0}^{2\pi }\sin 2\theta d\theta

=-\frac{\mu _{0}}{4\pi}\frac{ M}{2r} \int_{0}^{2\pi }\left ( -\frac{\cos 2 \theta }{2} \right )
=-\frac{\mu _{0}}{4\pi}\frac{ M}{2r}[\cos 4 \pi - \cos 0]=-\frac{\mu _{0}}{4\pi}\frac{ M}{2r} [1-1]=0

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