Question

 A magnetic dipole ~m = mˆk is placed in an otherwise uniform external magnetic

field B~

ex = −B0

ˆk .

(a) Write an expression for the radial, polar and azimuthal components of magnetic

field B~ (r, θ, φ) due to ~m only . (1 mark)

(b) Show that in the combined field there exists a certain spherical surface centered

on the dipole through which no field lines pass, Find its radius. (2 marks)

(c) Sketch the magnetic field lines in the region around the dipole in presence of

B~

ex.

Answer 1

Given,

Magnetic dipole $\overrightarrow{m}=m\hat{k}$

Uniform external magnetic field $=-B_o\hat{k}$

Net magnetic field $\overrightarrow{B}=B_o\hat{z}-\frac{\mu_o m_o }{4\pi r^3 }(2\cos\theta \hat{r}+\sin\theta \hat{\theta})$

For the radial component,

$\overrightarrow{B}.\hat{r}=0$

$\Rightarrow B_o\cos\theta -\frac{\mu_o m_o}{2 \pi r^3}\cos \theta=0$

After solving,

$r^3= [\frac{\mu_o m_o}{2 \pi B_o}]$

$r= [\frac{\mu_o m_o}{2 \pi B_o}]^{1/3}$

The radial component of the field is zero, is shows that no field of lines penetrate the spherical field.