Medium
$A$ sample of paramagnetic salt contains $2.0 \times 10^{24}$ atomic dipoles each of dipole moment $1.5 \times 10^{-23} \; J \, T^{-1}$. The sample is placed under a homogeneous magnetic field of $0.64 \; T$ and cooled to a temperature of $4.2 \; K$. The degree of magnetic saturation achieved is $15 \%$. What is the total dipole moment of the sample for a magnetic field of $0.98 \; T$ and a temperature of $2.8 \; K$? (Assume Curie's law)
(D) Number of atomic dipoles,$n = 2.0 \times 10^{24}$.
Dipole moment of each atomic dipole,$M = 1.5 \times 10^{-23} \; J \, T^{-1}$.
Total dipole moment of the sample if fully saturated,$M_{\text{total}} = n \times M = 2.0 \times 10^{24} \times 1.5 \times 10^{-23} = 30 \; J \, T^{-1}$.
Given saturation at $B_1 = 0.64 \; T$ and $T_1 = 4.2 \; K$ is $15 \%$.
Effective dipole moment $M_1 = 0.15 \times 30 = 4.5 \; J \, T^{-1}$.
According to Curie's law,the magnetization $M \propto \frac{B}{T}$,so $\frac{M_2}{M_1} = \frac{B_2}{B_1} \times \frac{T_1}{T_2}$.
Substituting the values for $B_2 = 0.98 \; T$ and $T_2 = 2.8 \; K$:
$M_2 = M_1 \times \frac{B_2}{B_1} \times \frac{T_1}{T_2} = 4.5 \times \frac{0.98}{0.64} \times \frac{4.2}{2.8}$.
$M_2 = 4.5 \times 1.53125 \times 1.5 = 10.336 \; J \, T^{-1}$.
Dipole moment of each atomic dipole,$M = 1.5 \times 10^{-23} \; J \, T^{-1}$.
Total dipole moment of the sample if fully saturated,$M_{\text{total}} = n \times M = 2.0 \times 10^{24} \times 1.5 \times 10^{-23} = 30 \; J \, T^{-1}$.
Given saturation at $B_1 = 0.64 \; T$ and $T_1 = 4.2 \; K$ is $15 \%$.
Effective dipole moment $M_1 = 0.15 \times 30 = 4.5 \; J \, T^{-1}$.
According to Curie's law,the magnetization $M \propto \frac{B}{T}$,so $\frac{M_2}{M_1} = \frac{B_2}{B_1} \times \frac{T_1}{T_2}$.
Substituting the values for $B_2 = 0.98 \; T$ and $T_2 = 2.8 \; K$:
$M_2 = M_1 \times \frac{B_2}{B_1} \times \frac{T_1}{T_2} = 4.5 \times \frac{0.98}{0.64} \times \frac{4.2}{2.8}$.
$M_2 = 4.5 \times 1.53125 \times 1.5 = 10.336 \; J \, T^{-1}$.
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Similar Questions
Match List-$I$ with List-$II$.
List-$I$ (Material) List-$II$ (Susceptibility $(\chi)$) $A$. Diamagnetic $I$. $\chi=0$ $B$. Ferromagnetic $II$. $0 > \chi \geq -1$ $C$. Paramagnetic $III$. $\chi >> 1$ $D$. Non-magnetic $IV$. $0 < \chi < \varepsilon$ (a small positive number)
Choose the correct answer from the options given below:
| List-$I$ (Material) | List-$II$ (Susceptibility $(\chi)$) |
| $A$. Diamagnetic | $I$. $\chi=0$ |
| $B$. Ferromagnetic | $II$. $0 > \chi \geq -1$ |
| $C$. Paramagnetic | $III$. $\chi >> 1$ |
| $D$. Non-magnetic | $IV$. $0 < \chi < \varepsilon$ (a small positive number) |
Choose the correct answer from the options given below:
MediumNEET 2024
View SolutionThe magnetic susceptibility of a paramagnetic material at $-73^{\circ} C$ is $0.0075$. Its value at $-173^{\circ} C$ will be
$A$ diamagnetic material in a magnetic field moves:
Electrical resistance of certain materials,known as superconductors,changes abruptly from a nonzero value to zero as their temperature is lowered below a critical temperature $T_c(0)$. An interesting property of superconductors is that their critical temperature becomes smaller than $T_c(0)$ if they are placed in a magnetic field,i.e.,the critical temperature $T_c(B)$ is a function of the magnetic field strength $B$. The dependence of $T_c(B)$ on $B$ is shown in the figure.
$1.$ In the graphs below,the resistance $R$ of a superconductor is shown as a function of its temperature $T$ for two different magnetic fields $B_1$ (solid line) and $B_2$ (dashed line). If $B_2 > B_1$,which of the following graphs shows the correct variation of $R$ with $T$ in these fields?
$2.$ $A$ superconductor has $T_c(0) = 100 \ K$. When a magnetic field of $7.5 \ T$ is applied,its $T_c$ decreases to $75 \ K$. For this material,one can definitely say that when:
$(A)$ $B = 5 \ T, T_c(B) = 80 \ K$
$(B)$ $B = 5 \ T, 75 \ K < T_c(B) < 100 \ K$
$(C)$ $B = 10 \ T, 75 \ K < T_c < 100 \ K$
$(D)$ $B = 10 \ T, T_c = 70 \ K$
Give the answer for questions $1$ and $2$.
$1.$ In the graphs below,the resistance $R$ of a superconductor is shown as a function of its temperature $T$ for two different magnetic fields $B_1$ (solid line) and $B_2$ (dashed line). If $B_2 > B_1$,which of the following graphs shows the correct variation of $R$ with $T$ in these fields?
$2.$ $A$ superconductor has $T_c(0) = 100 \ K$. When a magnetic field of $7.5 \ T$ is applied,its $T_c$ decreases to $75 \ K$. For this material,one can definitely say that when:
$(A)$ $B = 5 \ T, T_c(B) = 80 \ K$
$(B)$ $B = 5 \ T, 75 \ K < T_c(B) < 100 \ K$
$(C)$ $B = 10 \ T, 75 \ K < T_c < 100 \ K$
$(D)$ $B = 10 \ T, T_c = 70 \ K$
Give the answer for questions $1$ and $2$.
What is a magnetic substance? Write its types.
Medium
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