Question: Q. 6. (a) An iron ring of relative permeability $\mu_{r}$ has windings of insulated copper wire of $n$ turns per metre. When the current in the windings is $I$, find the expression for the magnetic field in the ring.
(b) Thesusceptibility of a magnetic material is 0.9853 . Identify the type of magnetic material. Draw the modification of the field pattern on keeping a piece of this material in a uniform magnetic field.
R [Delhi & OD, 2018]
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Solution:
Ans. (a) Expression for Ampere’s circuital law $1 / 2$
Derivation of magnetic field inside the ring
$1 / 2$
(b) Identification of the material
$1 / 2$
Drawing the modification of the field pattern $1 / 2$
(a) From Ampere’s circuital law, we have,
$$ \begin{equation*} \oint \vec{B} \cdot d \vec{l}=\mu_{0} \mu_{r} I_{\text {enclosed }} \tag{i} \end{equation*} $$
For the field inside the ring, we can write
$$ \begin{aligned} & \oint \vec{B} \cdot d \vec{l}=\oint B d l=B \cdot 2 \pi r \ & \text { ( } r=\text { radius of the ring) } \ & \text { Also, } \quad I_{\text {enclosed }}=(2 \pi r n) I \end{aligned} $$
using equation (i)
$\therefore \quad B \cdot 2 \pi r=\mu_{0} \mu_{r}(2 \pi r n) I$
$\therefore \quad B=\mu_{0} \mu_{r} n I$
[Award these $\left(\frac{1}{2}+\frac{1}{2}\right)$ marks even if the result is
written without giving the derivation] $1 / 2$
(b) The material is paramagnetic. $1 / 2$
The field pattern gets modified as shown in the figure below.
$1 / 2$
[CBSE Marking Scheme 2018]
Detailed Answer :
(a)
Apply Ampere’s Law for the magnetic field due to iron ring wounded by insulating copper wire, having current I, $\oint \bar{B} \cdot \overline{d l}=\mu^{\prime} \times($ current enclosed by closed path $) \quad 1 / 2$
or, $B d l \cos 0^{\circ}=\mu^{\prime} \times(n \times 2 \times r) \times I$
or, $B \times 2 \times r \times 1=\mu^{\prime} n \times 2 \times r \times I$
or, $\quad B=\mu^{\prime} n I$
But $\quad \mu_{r}=\frac{\mu^{\prime}}{\mu_{\mathrm{o}}}$
So, $\quad B=\mu_{0} \mu_{r} n I$
This is the required expression for magnetic field. Where, $\mu_{r} \rightarrow$ relative permeability, $\mu_{0} \rightarrow$ permeability of free space, $n \rightarrow$ number of turns per unit length $n$
(b) Given : susceptibility $\chi=0.9853$ since susceptibility $\chi$ given is + ve and less than unity i.e. $\chi<+1 \quad 1 / 2$ $\Rightarrow$ magnetic material is paramagnetic material. Thus when paramagnetic material is placed in the uniform magnetic field then the modified magnetic field is shown in figure.
Paramagnetic material