Magnetic Properties of Matter

The relationship between magnetic susceptibility $(\chi)$ and magnetic permeability $(\mu)$ is expressed as follows: Relative Permeability ($\mu_r$): This is defined as the ratio of the permeability of a material ($\mu$) to the permeability of free space ($\mu_0$).

$\mu = \mu_0 \mu_r \Rightarrow \mu_r = \dfrac{\mu}{\mu_0}$ Magnetic Susceptibility ($\chi$): This relates to relative permeability as follows: $\mu_r = 1 + \chi \Rightarrow \chi = \mu_r - 1$ Substituting the expression for $\mu_r$ from above: $\chi = \left(\dfrac{\mu}{\mu_0} - 1\right)$ Thus, the equation shows how magnetic susceptibility is derived from the magnetic permeability relative to free space.

The percentage change in the magnetic field (B) when the space within a current-carrying solenoid is filled with magnesium is calculated as follows: We start by using the formula for percentage change in the magnetic field: $\% \text{ change in } B = \dfrac{B_{\text{new}} - B_{\text{old}}}{B_{\text{old}}} \times 100\%$ Substituting the expressions for the magnetic field, we have: $= \dfrac{\mu n i - \mu_0 n i}{\mu_0 n i} \times 100\% = \dfrac{\mu - \mu_0}{\mu_0} \times 100\%$ Where: $\mu$ is the permeability of the material inserted (here, magnesium). $\mu_0$ is the permeability of free space. $n$ is the number of turns per unit length of the solenoid. $i$ is the current through the solenoid.

Given that the relationship between the permeability of the medium ($\mu$) and relative permeability ($\mu_r$) is: $\mu = \mu_0 \mu_r$ We can simplify this to: $= \dfrac{\mu_0 \mu_r - \mu_0}{\mu_0} \times 100\%$ Which reduces to: $= (\mu_r - 1) \times 100\%$ The relative permeability $\mu_r$ is equal to $1 + \chi$, where $\chi$ is the magnetic susceptibility.

Thus, we have: $= \chi \times 100\%$ When magnesium is used, with a magnetic susceptibility $\chi_{\text{Mg}} = 1.2 \times 10^{-5}$, the calculation becomes: $= 1.2 \times 10^{-5} \times 100\% = 1.2 \times 10^{-3}\%$ Therefore, the percentage increase in the magnetic field when the solenoid is filled with magnesium is $1.2 \times 10^{-3}\%$.

The assertion states that magnetic monopoles do not exist.

This is consistent with current experimental findings and the standard formulation of Maxwell's equations.

The reason provided is that magnetic field lines are continuous and form closed loops.

This is mathematically expressed by the Maxwell equation:

This equation tells us that there are no "sources" or "sinks" of magnetic fields (unlike electric fields, which can originate or terminate at charges).

In other words, magnetic field lines neither begin nor end; they loop continuously.

Since the fact that the magnetic field lines form closed loops is a direct consequence of the absence of isolated magnetic charges (monopoles), the reason correctly explains the assertion.

Thus, both (A) and (R) are correct, and (R) is the correct explanation of (A).

The most appropriate answer is Option A.

To determine the potential energy of a magnetic dipole in a uniform magnetic field, we start with the torque equation: $\tau = M \times B = MB \sin 60^\circ$ Given that $\tau = 80 \sqrt{3} \, \text{N m}$ and $\sin 60^\circ = \dfrac{\sqrt{3}}{2}$, we can equate and solve for $MB$: $MB \dfrac{\sqrt{3}}{2} = 80 \sqrt{3}$ Solving for $MB$, we find: $MB = 160$ The potential energy $U$ of the dipole in the magnetic field is given by: $U = -M \cdot B = -MB \cos 60^\circ$ Substituting in $\cos 60^\circ = \dfrac{1}{2}$, we calculate: $U = -160 \times \dfrac{1}{2} = -80 \, \text{J}$ Thus, the potential energy of the dipole is $-80 \, \text{J}$.

Coercivity, H =

Inside solenoid the magnetic field, B = $\mu$0ni $\therefore$ H =

= ni =

=

= 2600 A/m