Class 12 Physics Chapter 5 Magnetism and Matter

Key Concept: Gauss’s Law for Magnetism

c) Zero.
[Solution Description]
Gauss’s law for magnetism states that the total magnetic flux through a closed surface is zero because magnetic field lines form continuous closed loops, with no beginning or end. This means that whatever flux enters a volume also exits it, resulting in a net flux of zero.

Key Concept: Equatorial Field

a) $-\frac{\mu_0 m}{4 \pi r^3}$

[Solution Description] The formula for the equatorial field of a magnetic dipole as per the syllabus is given by:
$-\frac{\mu_0 m}{4 \pi r^3}$
where $\mu_0$ is the permeability of free space, m is the magnetic dipole moment, and r is the distance from the center of the dipole.

Key Concept: Advanced Formula Application, Complex Material Interaction

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

To solve this, we note that when a ferromagnetic material is placed inside a solenoid, the total magnetic field B inside the solenoid becomes significantly larger compared to when it contains just air or vacuum. This is due to the additional magnetic field $B_m$, which results from the magnetisation M of the ferromagnetic material.

According to the formula:

$B = B_0 + B_m = B_0 + \mu_0 M$

Here, $B_0$ is the magnetic field without any material core, and $B_m$ is the enhancement provided by the ferromagnetic material. Since ferromagnetic materials have high susceptibility $(\chi)$, the relationship $\mu_r = 1 + \chi$ implies that $\mu_r \gg 1$. Therefore, M is significantly higher in ferromagnetic materials, boosting $B_m$ substantially.

Therefore, both the Assertion and Reason are true, and the Reason correctly explains why the magnetic field inside the solenoid increases with a ferromagnetic core: because $\mu_r$ being much larger than 1 implies significant magnetisation properties that enhance the field.

Key Concept: Temperature Effect

b) Paramagnetic substances
[Solution Description] At elevated temperatures, ferromagnetic materials undergo a transition and become paramagnetic. This is due to increased thermal energy that disrupts the alignment of magnetic moments within the material.

Key Concept: Advanced Calculations, Domain Theory

b) $400000 A/m$
[Solution Description] To find the magnetization $M$, we use the formula $M = \frac{B – \mu_0 H}{\mu_0}$, where $B = \mu_r \mu_0 H$.

First, calculate $B$:
$B = \mu_r \mu_0 H = 500 \times 4\pi \times 10^{-7} \, \text{T m/A} \times 800 \, \text{A/m}.$
Simplify and compute this:
$B = 500 \times 4\pi \times 10^{-7} \times 800 = 0.5026548 \, \text{T}.$

Now, use the magnetization formula:
$M = \frac{B – \mu_0 H}{\mu_0} = \frac{0.5026548 – 4\pi \times 10^{-7} \times 800}{4\pi \times 10^{-7}}.$

Calculate:
$M = \frac{0.5026548 – 0.0010053}{4\pi \times 10^{-7}} \approx 399999 \, \text{A/m}.$
Therefore, the magnetization $M$ is approximately $400000 \, \text{A/m}$.

Key Concept: Field Enhancement

b) Field lines are concentrated inside
[Solution Description] In paramagnetic materials, the magnetic field lines become concentrated inside the material. This occurs because the individual atomic dipoles align partially with the external field, enhancing it slightly. Unlike diamagnetic substances, where the field is expelled, paramagnetic materials draw more field lines into themselves.

Key Concept: Magnetic Susceptibility

c) They have positive and small susceptibility.
[Solution Description] Paramagnetic materials have a small and positive magnetic susceptibility $(\chi)$. In these materials, the alignment of atomic dipoles occurs only when an external magnetic field is applied, contributing to a weak attraction to magnetic fields.

Key Concept: Temperature Effects

b) It becomes paramagnetic
[Solution Description] At high temperatures, often above the Curie temperature, ferromagnetic materials lose their strong magnetization properties and become paramagnetic. This happens because the thermal agitation disrupts the alignment of domains that contribute to ferromagnetism.

Key Concept: Torque in Fields

c) $p \times E$
[Solution Description]
The torque $\tau$ on an electric dipole in a uniform electric field is given by the expression:
$\tau = p \times E,$
where p is the electric dipole moment and E is the external electric field.

This results in a rotational effect that aligns the dipole with the electric field.

Key Concept: Basic Concept

c) $B_A = \frac{\mu_0}{4\pi} \frac{2m}{r^3}$
[Solution Description] The formula for the axial magnetic field $(B_A)$ of a bar magnet at a distance r where $(r >> l)$ is given by:
$B_A = \frac{\mu_0}{4\pi} \frac{2m}{r^3}$
This formula expresses the axial field in terms of permeability of free space $(\mu_0)$, the magnetic moment (m), and the distance (r).

Key Concept: Potential Energy Variation

a) $0^\circ$
[Solution Description]
The potential energy $U_m$ of a magnetic dipole in a uniform magnetic field is given by:
$U_m = -mB\cos\theta$

Potential energy is minimum when $\cos\theta$ is maximum. Since $\cos\theta$ ranges from -1 to 1, it is maximum at $\theta = 0^\circ$.

Thus, the potential energy is minimum when $\theta = 0^\circ$.

Key Concept: Magnetic Equations

c) $4.4 \times 10^{-5} \, \text{T}$
[Solution Description] To find the magnetic field B, we use the equation $B = \mu_0 (H + M)$.

First, substitute the given values into the equation:
$B = 4\pi \times 10^{-7} \times (200 + 150).$

Calculate inside the parentheses:
$200 + 150 = 350.$

Now multiply:
$B = 4\pi \times 10^{-7} \times 350.$

Since $\pi \approx 3.1416$,
$B = 4 \times 3.1416 \times 10^{-7} \times 350.$

Calculating further:
$B \approx 1.25664 \times 10^{-7} \times 350.$

Therefore,
$B \approx 4.39824 \times 10^{-5} \, \text{T}.$

The closest approximate value considering significant figures is $4.4 \times 10^{-5} \, \text{T}$.

Key Concept: Real-World Application

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

– Superconductors exhibit a phenomenon called the Meissner effect, where they expel all magnetic flux from their interior. This results in magnetic susceptibility $\chi = -1$, meaning they oppose an applied magnetic field so strongly that the net internal magnetic field (B) becomes zero.

– Since $\chi = -1$, relative permeability $\mu_r = 1 + \chi = 0$, which leads to $\mu = \mu_0 \cdot \mu_r = 0$.

– Therefore, both the assertion and the reason are true, and the reason correctly explains why superconductors have $\mu = 0$ due to complete field expulsion.

Key Concept: Intersection Rule

b) The direction of the magnetic field would not be unique
[Solution Description] If magnetic field lines were to intersect, it would imply that the direction of the net magnetic field is not unique at the point of intersection, which contradicts the properties of a field line.

Key Concept: Cause and Effect

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

The assertion states that increasing the temperature of a ferromagnetic material reduces its magnetic susceptibility. This is true because as temperature increases, thermal energy disrupts the alignment of magnetic domains in ferromagnetic materials, leading to a decrease in their magnetic susceptibility.

The reason explains why this happens: thermal agitation at higher temperatures causes more random motion and disorder among the magnetic domains, reducing their ability to align uniformly under an external magnetic field.

Since both the assertion and the reason are correct and the reason correctly explains the phenomenon described in the assertion, the correct answer is option (a).

Key Concept: Domain Growth

b) They grow in size
[Solution Description] When an external magnetic field is applied to a ferromagnetic material, the domains that are aligned with the field grow in size. This occurs as the domain walls move to increase the alignment with the field direction, reducing the overall energy of the system.

Key Concept: Calculation of Magnetic Field

b) $6.28 \times 10^{-3} \, T$

[Solution Description] To calculate the magnetic field B using the given formula:

$B = \mu_r \mu_0 H$

where $\mu_r = 5$, $H = 1000 \, A/m$, and $\mu_0 = 4\pi \times 10^{-7} \, T\cdot m/A$.

Substituting these values:

$B = 5 \times 4\pi \times 10^{-7} \times 1000$

$B = 20\pi \times 10^{-4} \, T$

$B \approx 6.28 \times 10^{-3} \, T$

Therefore, the magnetic field B is approximately $6.28 \times 10^{-3} \, T$.

Key Concept: Magnetic Field Visualization

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

The assertion states that iron filings around a bar magnet form a pattern, which is true. This pattern represents the invisible magnetic field lines. The reason explains that these lines are indeed continuous and form closed loops, which is also true. The reason correctly explains why the iron filings form such a pattern – they align along the magnetic field lines due to their susceptibility to magnetic forces, illustrating the continuity of the lines.

Key Concept: Compass Needle

c) Compass needle
[Solution Description] A compass needle aligns itself along a magnetic field line due to Earth’s magnetic field and can be used to find the local direction of the magnetic field.

Key Concept: Formula Application

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description] To verify the Assertion, we calculate the magnetic field B using the formula $B = \mu_r \mu_0 H$. Given: $\mu_r = 2$, $H = 500 \, \text{A/m}$, and $\mu_0 = 4\pi \times 10^{-7} \, \text{T m/A}$.

Let’s substitute these values into the formula:

$B = 2 \times 4\pi \times 10^{-7} \times 500$

First calculate the multiplication inside the equation:

$B = 2 \times (4\pi \times 10^{-7}) \times 500$

Calculate $4\pi \approx 12.5663706144$

$B = 2 \times 12.5663706144 \times 10^{-7} \times 500$

Now multiply $[12.5663706144 \times 10^{-7} \times 500 \approx 6.2831853072 \times 10^{-4}]$

$[B = 2 \times 6.2831853072 \times 10^{-4} \, \mathrm{T}]$

Finally multiply by 2 to get:

$[B = 1.25663706144 \times 10^{-3} \approx 1.26 \times 10^{-3} \, \mathrm{T}]$

Therefore, both the assertion and the reason are true, and the reason correctly explains the assertion.

Key Concept: Domain Alignment

c) The domains align with the external field.
[Solution Description] In a ferromagnetic material, the atoms are grouped in regions called domains, where their magnetic moments are aligned. When an external magnetic field is applied, these domains tend to align themselves along the direction of the field. This causes an increase in the overall magnetization of the material as more domains align and grow in size.

Key Concept: Field Line Direction

c) The direction of the net magnetic field B
[Solution Description] The tangent to a magnetic field line at any given point provides the direction of the net magnetic field B at that point. This concept is similar to electric field lines where the tangent gives the electric field direction.

Key Concept: Relative Permeability

b) $\mu_r = 1 + \chi$
[Solution Description] Relative permeability $\mu_r$ is the sum of 1 and magnetic susceptibility $\chi$. The relationship is mathematically expressed as:
$\mu_r = 1 + \chi$

Key Concept: Field Line Continuity

b) Continuous
[Solution Description] The magnetic field lines around both a bar magnet and a solenoid are continuous loops. The lines emerge from the north pole (or end) and enter the south pole, maintaining continuity throughout their path.

Key Concept: Historical Magnetism

c) From the Greek island of Magnesia
[Solution Description] The term “magnet” originates from the Greek island of Magnesia. It was named after this location due to the natural occurrence of lodestones, which were found to attract iron.

Key Concept: Equation Application

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

[Solution Description] In this question, we need to analyze both the assertion and reason separately.

For the Assertion (A), it states that magnetic susceptibility $\chi$ is defined as the ratio of magnetisation M to magnetic intensity H:

$\chi = \frac{M}{H}$

This definition aligns with the concept of magnetic susceptibility, which measures how easily a material can be magnetized. Therefore, Assertion (A) is true.

Now, for the Reason (R), it mentions the net magnetic field inside a solenoid when a material is present:

$B = B_0 + B_m = B_0 + \mu_0 M$

Here, $B_0 = \mu_0 nI$ is the initial magnetic field produced by the current and $B_m = \mu_0 M$ is the additional field contributed by the material inside the solenoid. This equation is indeed true and accurately explains the total magnetic field within the solenoid when a core material is introduced. Hence, Reason (R) is also correct.

Although both statements are true, the Reason does not directly explain the concept presented in the Assertion about magnetic susceptibility $\chi$. Hence, they are two independent correct statements.

Thus, the correct answer is (b).

Key Concept: Gauss’s Law of Magnetism

c) It is always zero.
[Solution Description] Gauss’s law of magnetism states that the net magnetic flux through any closed surface is zero. This is because magnetic monopoles do not exist; hence, every magnetic line that enters a surface must also exit it.

Key Concept: Field Line Continuity

d) Magnetic field lines emerge from the north pole and enter the south pole
[Solution Description] Magnetic field lines remain continuous in a bar magnet, emerging from one face (north pole) and entering the other (south pole), forming closed loops without beginning or end.

Key Concept: Magnetic Permeability

c) Greater than 1000
[Solution Description] The relative magnetic permeability of ferromagnetic materials is significantly greater than 1000, which allows them to be strongly attracted to magnetic fields.

Key Concept: Magnetic Moment Analogy, Energy and Stability

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

[Solution Description]

To solve this problem, we must analyze both statements separately:

– **Assertion**: The torque $\tau$ experienced by a bar magnet in a uniform magnetic field B is given by the vector cross product formula $\tau = m \times B$, where $m$ is the magnetic moment of the bar magnet. The magnitude of this torque can be expressed as $\tau = mB\sin\theta$, where $\theta$ is the angle between m and B. The sine function reaches its maximum value of 1 when $\theta = 90^\circ$, making the torque maximum at this orientation. Thus, the assertion is true.

– **Reason**: The potential energy U of a bar magnet in a magnetic field is given by the dot product $U = -m.B = -mB\cos\theta$. The cosine function achieves its maximum value of 1 when $\theta = 0^\circ$, meaning m is parallel to B. In this scenario, U attains its minimum value since it involves a negative sign. Therefore, the reason is correct.

Both statements are true; however, the reason does not directly explain why the torque is maximized, as they address different aspects of the bar magnet’s behavior in a magnetic field. Therefore, the correct answer is option b).

Key Concept: Advanced Flux Analysis, Real-world Application

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description] In this problem, we need to consider both the assertion and the reason separately and then determine their relationship. The assertion states that the net magnetic flux through a closed surface surrounding an electromagnet with varying current is zero. This is true because Gauss’s law for magnetism states that the net magnetic flux through any closed surface is indeed zero, irrespective of the changing nature of the current in the electromagnet. Now, reasoning provided correctly explains why the assertion holds true since it directly refers to Gauss’s Law for magnetism. Therefore, both the assertion and reason are correct, with reason being the correct explanation for the assertion.

Key Concept: Field Line Analysis

d) Assertion is false, but Reason is true.

[Solution Description]

The assertion states that magnetic field lines between two magnetic poles are perfectly straight, which implies an idealized situation without any disturbances or variations at the edges. However, this is not true in practical situations. Due to the nature of magnetic fields and the presence of magnetic dipoles, some fringing of the lines occurs around the edges where the field spreads out slightly before converging again. This phenomenon is a consequence of the fact that magnetic field lines form closed loops and must maintain continuity. Hence, the reason provided is correct as it explains why perfect straightness cannot be maintained. Therefore, (A) is false but (R) is true.

Key Concept: Basic Concept

a) A Greek island

[Solution Description] The word ‘magnet’ originates from a Greek island named Magnesia, where magnetic stones were first discovered.

Key Concept: Formula Application

c) $U = -m \cdot B$
[Solution Description] The potential energy U of a magnetic dipole in a uniform magnetic field is given by the formula:
$U = -m \cdot B$
where m is the magnetic dipole moment and B is the magnetic field strength.

Key Concept: Weakness of Diamagnetism

b) Because it is weaker than other magnetic effects like paramagnetism and ferromagnetism
[Solution Description] Diamagnetism is present in all materials but usually gets overshadowed by stronger magnetic effects such as paramagnetism and ferromagnetism. This makes it hard to observe in everyday scenarios unless those other effects are absent or negligible.

Key Concept: Equatorial Field

b) $B_E = -\frac{\mu_0 m}{4\pi r^3}$
[Solution Description]
According to the syllabus, the formula for the equatorial field of a bar magnet is
$B_E = -\frac{\mu_0 m}{4\pi r^3}$

Key Concept: Real-World Application

c) Transformers
[Solution Description] Soft ferromagnetic materials are used in applications where temporary magnetization is needed, such as in transformers and electromagnets. These materials lose their magnetization once the external magnetic field is removed, making them ideal for use in electrical devices that require fluctuating magnetic fields. The characteristic that makes soft ferromagnetic materials suitable for these applications is their low coercivity, which allows them to easily gain and lose magnetization. Hence, the correct choice reflects this usage in transformers.

Key Concept: Weakness of Effect

c) It is overshadowed by other magnetic effects

[Solution Description] Diamagnetism is a weak effect present in all substances but is frequently overshadowed by other more prominent magnetic effects like paramagnetism or ferromagnetism. Thus, it is not easily observed.

Key Concept: Magnetic Field Interaction

b) From weak to strong field

[Solution Description] Paramagnetic substances tend to move from regions of weak magnetic fields to regions of strong magnetic fields. This behavior is due to the alignment of permanent dipole moments in such substances with the external magnetic field.

Key Concept: Magnetic Field Lines

b) Magnetic field lines can intersect with electric field lines
[Solution Description] Magnetic field lines are used to represent the presence and direction of a magnetic field. They do not cross each other, as crossing would imply two directions for the same point in space, which is impossible. Furthermore, they form continuous closed loops without beginning or end since there are no magnetic monopoles.

Key Concept: Magnetic Field Calculation

b) $B = \frac{\mu_0}{4\pi} \frac{m}{r^3}$
[Solution Description] The formula for the magnetic field B at a point on the equatorial line of a long bar magnet is given by:

$B = \frac{\mu_0}{4\pi} \frac{m}{r^3}$

Where:
– $\mu_0$ is the permeability of free space,
– m is the magnetic moment,
– r is the distance from the center of the magnet.

Given that $r \gg l$, the simplified and applicable equation becomes $B = \frac{\mu_0}{4\pi} \frac{m}{r^3}$.

Key Concept: Field Line Patterns

d) They form continuous closed loops around a magnet.
[Solution Description] Magnetic field lines always form continuous closed loops because they do not have a start or end point, unlike electric field lines which begin on positive charges and end on negative charges.

Key Concept: Advanced Torque Calculation, Potential Energy Variation

c) Assertion is true, but Reason is false.

[Solution Description] The torque $\tau$ on a magnetic dipole is given by the formula $\tau = mB \sin \theta$, where $\theta$ is the angle between the magnetic moment $m$ and the magnetic field $B$. Initially, when $\theta = 90^\circ$, $\sin \theta = 1$; hence, the initial torque $\tau_1 = mB \cdot 1 = mB$. When the angle changes to $60^\circ$, $\sin \theta = \frac{\sqrt{3}}{2}$; thus, the new torque $\tau_2 = mB \cdot \frac{\sqrt{3}}{2} < mB$. This confirms that the torque decreases as the angle reduces from $90^\circ$ to $60^\circ$. For potential energy, the formula is given by $U = -mB \cos \theta$. At $90^\circ$, $U_1 = -mB \cdot 0 = 0$. At $60^\circ$, $U_2 = -mB \cdot \frac{1}{2} = -\frac{mB}{2}$, which means the potential energy actually decreases (becomes more negative) as the angle decreases from $90^\circ$ to $60^\circ$. Therefore, the reason statement about potential energy increasing is incorrect.