Class 12 Physics Chapter 6 Electromagnetic Induction

Key Concept: Formula Application

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

[Solution Description]

The assertion statement correctly presents the formula for mutual inductance, which involves $$\mu_0$$ (the permeability of free space), $$n_1$$ and $$n_2$$ (the turn densities of the solenoids), and geometric factors like $$r_1$$ (radius of the inner solenoid) and l (common length). The reason also accurately describes one of the assumptions made while deriving this formula, which is the assumption of a uniform magnetic field to simplify calculations by neglecting edge effects. Thus, both statements are true, and the reason correctly explains the assertion.

Key Concept: Critical Analysis, Design Consideration

c) Improve the cooling system
[Solution Description]
To optimize the heating process within the given frequency limits, consider these factors:

– **Coil Turns**: Increase the number of turns if possible; this helps in increasing the inductance and reducing current for the same power, thereby reducing heat generation in the coil.
– **Cooling System**: Enhance the cooling system to better dissipate heat, allowing the coil to safely operate closer to the upper frequency limit.
– **Material Selection**: Use materials with higher thermal conductivity for the coil to efficiently pass the heat away.
– **Impedance Matching**: Optimize impedance matching networks to ensure maximum power transfer without losses.

Option c involves enhancing the cooling system which directly allows safe operation close to 80 kHz.

Key Concept: Simple Calculation

d) -6 V
[Solution Description] The induced emf (E) in a solenoid can be calculated using the formula:
$$
E = -L \frac{di}{dt}
$$
where \$L\$ is the inductance (2 H), and $$\frac{di}{dt}$$ is the rate of change of current (3 A/s). Substituting these values:
$$
E = -(2\, \text{H}) \times (3\, \text{A/s}) = -6\, \text{V}
$$
The negative sign indicates the direction of the induced emf opposes the change in current.

Key Concept: Design Techniques

c) Assertion is true, but Reason is false.

[Solution Description]

The assertion is based on the principle that laminating the core reduces the path available for eddy currents, hence reducing their magnitude. However, the reason given states that lamination increases magnetic permeability, which is incorrect. Lamination primarily serves to increase the electrical resistivity along the plane perpendicular to the lamination and thereby reduce eddy current paths, not to increase the magnetic permeability.

Key Concept: Complex Calculation, Induced EMF Variation

c) 0.82 V
[Solution Description]
To find the induced emf, we first calculate the magnetic flux $$\Phi_B$$ through the loop. The area of the circle is given by $$A = \pi r^2 = \pi (0.1)^2 = 0.01\pi \text{ m}^2.$$

Magnetic flux $$\Phi_B$$ is given by:
$$\Phi_B = B(t) \cdot A = (4t^2 + 2t) \cdot 0.01\pi = (0.04\pi t^2 + 0.02\pi t)$$

According to Faraday’s law, the induced emf $$\varepsilon$$ is:
$$\varepsilon = -\frac{d\Phi_B}{dt} = -\frac{d}{dt}(0.04\pi t^2 + 0.02\pi t)$$

Calculating the derivative:
$$\frac{d\Phi_B}{dt} = 0.08\pi t + 0.02\pi$$

Substituting $$t = 3 s:$$
$$\varepsilon = -(0.08\pi \cdot 3 + 0.02\pi) = -(0.24\pi + 0.02\pi) = -0.26\pi$$

Therefore, the induced emf at $$t = 3$$ s is approximately $$-0.82 V$$ $$(since \pi \approx 3.14).$$

Key Concept: Basic Concept

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

[Solution Description] The formula $$ \varepsilon = Blv $$ represents the induced electromotive force (emf) in a conductor moving with velocity v perpendicular to a uniform magnetic field B , where l is the length of the conductor within the magnetic field. This relationship shows that both Assertion and Reason are true, and the Reason correctly explains why the induced emf occurs—due to the relative motion creating a change in the magnetic flux linked with the circuit formed by the moving conductor.

Key Concept: Simple Calculation

b) $$W = \frac{1}{2}LI^2$$
[Solution Description] The energy W stored in an inductor with inductance L and current I is given by the formula $$W = \frac{1}{2} LI^2$$.

Key Concept: Concept Application

c) $$0.16 \, V$$
[Solution Description] Use the formula for induced emf: $$\epsilon = -N \frac{d\Phi_B}{dt}$$, where $$N=1$$ since it’s a single loop. Calculate the initial and final magnetic flux:

$$
\Phi_{B_{\text{initial}}} = B_i \cdot A = 0.5 \times (0.4)^2 = 0.08 \, \text{Wb}
$$

$$
\Phi_{B_{\text{final}}} = B_f \cdot A = 0 \times (0.4)^2 = 0 \, \text{Wb}
$$

Find the rate of change of magnetic flux:

$$
\frac{d\Phi_B}{dt} = \frac{\Phi_{B_{\text{final}}} – \Phi_{B_{\text{initial}}}}{0.5} = \frac{0 – 0.08}{0.5} = -0.16 \, \text{Wb/s}
$$

Thus, the induced emf is:

$$
\epsilon = -1 \times (-0.16) = 0.16 \, \text{V}
$$

So, the induced emf is $$0.16 \, V$$.

Key Concept: Conceptual Application

a) 5.33 V
[Solution Description] The formula for the self-induced emf $$\epsilon$$ in a coil is given by $$\epsilon = -L \frac{dI}{dt}$$. Here, $$L = 4 \, \text{H}$$, the change in current $$\Delta I = 6 \, \text{A} – 2 \, \text{A} = 4 \, \text{A}$$, and the time $$\Delta t = 3 \, \text{s}$$. So, $$\frac{dI}{dt} = \frac{\Delta I}{\Delta t} = \frac{4}{3} \, \text{A/s}$$. Therefore, $$\epsilon = -4 \times \frac{4}{3} = -\frac{16}{3} \, \text{V}$$. The magnitude is $$\frac{16}{3} \, \text{V}$$.

Key Concept: Basic Units

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

[Solution Description] Both the assertion and reason are true, and they represent the same concept with different expressions for the unit of magnetic flux. In terms of basic SI units, 1 Weber is equivalent to 1 Tesla meter squared, thus both are correct ways to describe the unit of magnetic flux.

Key Concept: Complex Induced Emf Scenario, Advanced Lorentz Force Scenario

c) 8.94 mV
[Solution Description]
To solve this problem, we need to find the change in magnetic flux through the loop as it rotates and then use Faraday’s law to calculate the induced emf.

The area of the loop is:
$$A = 5 \times 10 = 50 \, \text{cm}^2 = 50 \times 10^{-4} \, \text{m}^2 = 5 \times 10^{-3} \, \text{m}^2$$

The initial magnetic flux through the coil can be determined considering that initially the plane of the loop is perpendicular to the magnetic field:
$$\Phi_{B_i} = B_i \cdot A \cdot \cos(\theta_i) = 0$$
Since the loop starts rotating from rest, $$\theta_i = 90^\circ$$, making the cosine term zero.

After rotating through $$45^\circ$$, the orientation of the loop changes, the new angle between magnetic field and area vector becomes $$\theta_f = 45^\circ$$. Therefore, the flux is:
$$\Phi_{B_f} = (kx \cdot l)(w \cdot h)\cos(45^\circ)$$

The rate of change of flux can be modeled as:
$$\frac{d\Phi_B}{dt} = \frac{\Phi_{B_f} – \Phi_{B_i}}{\Delta t}$$

Since the transition takes time related to its angular speed:
$$\Delta t = \frac{45^\circ}{\omega} = \frac{\pi/4}{100} = \frac{\pi}{400} \, \text{s}$$

Using Faraday’s Law for absolute value:
$$|\epsilon| = \left| – \frac{d\Phi_B}{dt} \right|$$

We substitute known values into the equation and get the result:
$$|\epsilon| = \left| – \frac{1 \times 10^{-3} \times 5 \times 10^{-3} \times \sin(45^\circ)}{\pi / 400} \right|$$

Simplifying further:
$$|\epsilon| = \frac{5 \times 10^{-6} \times \sqrt{2}/2}{\pi / 400}$$

Calculating the above expression gives:
$$|\epsilon| \approx 8.94 \, \text{mV}$$

Therefore, the magnitude of the induced emf is approximately $$8.94 \, \text{mV}.$$

Key Concept: Complex Cause Analysis, Eddy Currents and Energy Loss, Eddy Currents in Non-uniform Fields

c) Assertion is true, but Reason is false.

[Solution Description] Eddy currents are induced due to variations in magnetic flux, especially in non-uniform fields. Slotted conductors act to reduce these currents by disrupting continuous paths for current flow, thereby minimizing them rather than increasing energy loss. This misalignment between reduced eddy current formation and reason’s claim about increased dissipation results in a false assertion when considering the stated reason. Thus, the assertion is true but the provided reason for it is incorrect.

Key Concept: Units of Inductance

c) Henry
[Solution Description] The SI unit of self-inductance is the Henry (H), named after Joseph Henry, who discovered electromagnetic induction independently of Faraday.

Key Concept: Complex Scenarios, Advanced Calculations

b) 0.004 A
[Solution Description] According to Faraday’s Law of electromagnetic induction, the induced emf $$\varepsilon$$ can be calculated as:
$$
\varepsilon = – \frac{d\Phi}{dt}
$$
where $$\frac{d\Phi}{dt} = 0.02$$ Wb/s. Thus,
$$
\varepsilon = -0.02 \text{ V}
$$
Using Ohm’s Law, $$I = \frac{\varepsilon}{R}$$, where $$R = 5$$ ohms:
$$
I = \frac{-0.02}{5} = -0.004 \text{ A}
$$
The negative sign indicates direction opposed by Lenz’s law.

Key Concept: Complex Scenarios, Advanced Application

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

[Solution Description] According to Faraday’s Law of Induction, the induced emf $\epsilon$ in a coil is given by $$\epsilon = -N \frac{d\Phi_B}{dt}$$, where $\Phi_B$ is the magnetic flux through one turn of the coil. If a coil rotates in a non-uniform magnetic field, the rate of change of magnetic flux $d\Phi_B/dt$ will not be constant, as the field strength varies with position. Thus, $\epsilon$ would vary non-linearly with time. This directly correlates with the reason provided: the non-uniformity causes a non-linear change in flux, leading to a non-linear variation in induced emf.

Key Concept: Complex Calculation, Design Consideration

b) Double the area of the coil
[Solution Description]
Let’s denote the original magnetic field strength as $$ B_1 $$ and the original area of the coil as $$ A_1 = 0.25 \, \text{m}^2 $$.

The induced emf is given by:
$$
e_0 = NBA\omega
$$

Since we are reducing the magnetic field to half, the new magnetic field strength is $$ B_2 = \frac{B_1}{2} $$.

We want to keep the same peak emf, so:
$$
N B_1 A_1 \omega = N \left(\frac{B_1}{2}\right) A_2 \omega
$$

Solving for $$ A_2 $$:
$$
A_2 = 2A_1 = 2 \times 0.25 = 0.50 \, \text{m}^2
$$

Therefore, to maintain the same peak emf, the area of the coil must be doubled.

Key Concept: Conceptual Understanding

c) Inductance
[Solution Description]
In electromagnetic systems, inductance L is analogous to mass m in mechanical systems because both store energy; inductance stores magnetic potential energy, while mass stores kinetic energy.

Key Concept: Core Material

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

[Solution Description]

The assertion is true because the core material impacts the efficiency of a transformer by influencing hysteresis and eddy current losses. High permeability materials such as silicon steel are used to reduce these energy losses, thereby improving efficiency.

As for the reason, it is also true that laminating the core is a technique used to minimize eddy current losses. Eddy currents are loops of electric current induced within conductors by a changing magnetic field in the conductor, and they cause significant power loss. Lamination increases the resistance path of these currents, reducing the magnitude of the currents and hence the associated losses.

However, while both the assertion and reason are true individually, the reason given does not directly explain why the core material affects the transformer’s efficiency. Core material affects efficiency primarily through its permeability and ability to reduce hysteresis losses rather than just by being laminated.

Key Concept: Real-world Application

b) Thermal power
[Solution Description]
The conversion of steam energy into mechanical energy to drive the rotor shaft is typically associated with thermal power generation. In thermal power plants, coal, oil, or natural gas is burned to produce steam from water, which then drives turbines connected to generators.

Key Concept: Basic Concept

d) Emf generated by the motion of a conductor in a magnetic field
[Solution Description] Motional emf is the electromotive force induced when a conductor moves through a magnetic field, calculated by the formula $$\varepsilon = Blv,$$ where B is the magnetic field strength, l is the length of the conductor, and v is its velocity.

Key Concept: Basic Concept

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

[Solution Description]

The assertion states a formula for the energy stored in an inductor, which is correctly characterized by $$\frac{1}{2}LI^2$$, where L is the inductance and I is the current through the inductor. This equation describes how the energy is related to these parameters.

The reason mentions that inductance depends on the geometry of the coil and intrinsic material properties, such as permeability, which is also true. However, this reason does not explain why the specific formula for energy stored in an inductor is $$\frac{1}{2}LI^2$$

Therefore, both the assertion and the reason are true statements, but the reason is not the correct explanation of the assertion.

Key Concept: Basic Formula

b) $M = \mu_r \mu_0 n_1 n_2 \pi r_1^2 / l$
[Solution Description] The formula for mutual inductance M in the presence of a medium with relative permeability $\mu_r$ is given by:
$M = \mu_r \mu_0 n_1 n_2 \pi r_1^2 / l$
Here, $\mu_0$ is the permeability of free space, $n_1$ and $n_2$ are the number of turns in each coil, $r_1$ is the radius of one of the coils, and l is the length over which the coils overlap.

Key Concept: Problem Solving, Synthesis and Evaluation

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

[Solution Description] The assertion states that combining laminated cores with slotted rotors minimizes energy losses due to eddy currents. Laminated cores indeed increase the electrical resistance path for eddy currents by segmenting the core material into thin layers, thereby reducing the effective area in which these currents can circulate. This action increases the overall resistance encountered by eddy currents, thus minimizing them.

Slotted rotors are designed to disrupt the continuous paths available for eddy currents to form within the rotor structure. By introducing physical slots, the path for eddy current circulation is physically interrupted, leading to a reduction in their intensity.

Both Assertion and Reason are true, and they provide complementary actions towards minimizing eddy current losses, however, the reason alone does not fully explain the effectiveness of using both methods together as described in the assertion. Thus, the assertion is correct but the reason does not completely explain it.

Key Concept: Calculation-Based

c) 3 V

[Solution Description]

The formula for motional emf is given by $$e = B l v,$$ where B is the magnetic field, l is the length of the rod, and v is the velocity.

Using the values from the question:

$$e = 2 \, \text{T} \times 0.5 \, \text{m} \times 3 \, \text{m/s} = 3 \, \text{V}$$

Therefore, the motional emf induced in the rod is 3 V.

Key Concept: Effect of Coil Shape

c) Assertion is true, but Reason is false.

[Solution Description]

The assertion is true because changing the shape of a coil affects its area and orientation, which directly influences the magnetic flux according to the formula $$\Phi_B = B \cdot A \cdot \cos{\theta}$$, where B is the magnetic field strength, A is the area, and $$\theta$$ is the angle between the field and normal to the area. Altering any of these changes the flux. The reason provided is false because the induced emf is actually directly proportional to the number of turns in the coil, as per the formula $$\epsilon = -N\frac{d\Phi_B}{dt}$$. Therefore, increasing the number of turns increases the emf, not decreases it.

Key Concept: Function of Stator

c) Assertion is true, but Reason is false.

[Solution Description] The stator of an AC generator indeed contains the armature winding, which is responsible for generating electrical power as it rotates within a magnetic field. However, the rotor is the part that rotates, not the part that remains stationary. Thus, while the assertion is correct, the reason provided is false.

Key Concept: Rate of Work Done

c) $$I\epsilon$$
[Solution Description]
The formula for the rate of work done in a circuit with current I and emf $$\epsilon$$ is given by:
$$\frac{dW}{dt} = I\epsilon$$
This equation describes the power being input into the system as the current establishes itself against the back emf.

Key Concept: Open Circuit Emf

b) Emf is induced across the open ends

[Solution Description]

When a magnet is moved toward an open circuit, an electromotive force (emf) is induced across the open ends due to the change in magnetic flux through the circuit. However, since the circuit is open, no current flows despite the presence of emf.

Key Concept: Induction Factors

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

[Solution Description] According to Faraday’s law of electromagnetic induction, the induced emf $\varepsilon$ in a coil is given by $$\varepsilon = -N \frac{d\Phi}{dt}$$, where N is the number of turns and $\frac{d\Phi}{dt}$ is the rate of change of magnetic flux. Hence, increasing the number of turns (N) increases the value of $\varepsilon.$ Additionally, the statement about the rate of change of magnetic flux directly affecting the magnitude of induced emf is also true as per Faraday’s law. However, the reason does not specifically explain how increasing the number of turns affects the induced emf; it explains the relationship between emf and the rate of change of magnetic flux.

Key Concept: Direction of Current

b) Counter-clockwise
[Solution Description]
Lenz’s Law states that the direction of the induced current is such that it opposes the change in magnetic flux through the loop. If the loop is moved out of the magnetic field, the induced current will flow in a direction that attempts to maintain the original magnetic flux within the loop, essentially trying to induce a magnetic field inside the loop in the same direction as before.

Key Concept: Induced Emf Calculation

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

[Solution Description]

The assertion states that increasing the number of turns in a coil leads to an increase in the induced emf. This is true because the induced emf, according to Faraday’s law, is given by the rate of change of magnetic flux linkage, which is proportional to N, the number of turns. As N increases, the total flux linkage $$N\Phi_B$$ also increases, leading to a higher induced emf.

The reason provides the concept of flux linkage being equal to $$N\Phi_B$$ for a closely wound coil. This is true and directly relates to how the induced emf is calculated, as it shows the dependence on the number of turns.

Since both the assertion and the reason are true, and the reason correctly explains why the induced emf increases with more turns (by showing the relationship through flux linkage), the correct option is (a).

Key Concept: Intermediate Calculation

c) 400 N
[Solution Description]
To calculate the new braking force, we use the fact that the braking force (F) is proportional to the speed (v) of the conductor. Hence, we have:
$$
F_1 = k \cdot v_1
$$
$$
F_2 = k \cdot v_2
$$
Dividing these equations gives us:
$$
\frac{F_2}{F_1} = \frac{v_2}{v_1}
$$
Substituting the given values:
$$
\frac{F_2}{300} = \frac{20}{15}
$$
Solving for $$F_2:$$
$$
F_2 = 300 \times \frac{20}{15} = 400 \, \text{N}
$$
The new braking force is 400 N.

Key Concept: Coil Interaction

b) Deflection occurs momentarily and then stops

[Solution Description] When Coil C1 is powered, it generates a magnetic field. This changing magnetic field induces an electromotive force (emf) in Coil C2 due to electromagnetic induction. As a result, a current will be induced in Coil C2, which will cause a deflection in the galvanometer indicating that an emf has been induced in the second coil.

Key Concept: Basic Principle

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

[Solution Description]

The principle operation of an AC generator relies on electromagnetic induction, which occurs when the orientation of a loop within a magnetic field changes. This change in orientation alters the magnetic flux through the loop and induces an electromotive force (emf). Both the assertion and reason are true, and the reason correctly explains why the principle of an AC generator involves changing the loop’s orientation: this change leads to the induction of emf due to the variation in magnetic flux.

Key Concept: Definition Recall

d) $$\Phi_B = B \cdot A = BA \cos \theta$$

[Solution Description]

The magnetic flux $$\Phi_B$$ through a surface is given by the formula $$\Phi_B = B \cdot A = BA \cos \theta$$, where B is the magnetic field, A is the area of the surface, and $$\theta$$ is the angle between the magnetic field and the normal to the surface.

So, the correct answer is $$\Phi_B = B \cdot A = BA \cos \theta$$.

Key Concept: AC Generator Mechanics

d) Assertion is false, but Reason is true.

[Solution Description]

– In an AC generator, the induced emf is given by $$e = NBA \omega \sin \omega t$$ The term $$\sin \omega t$$ reaches its maximum value of 1 when $$\omega t = \frac{\pi}{2}$$ or any odd multiple of $$\frac{\pi}{2}$$ This condition corresponds to the plane of the coil being parallel to the magnetic field lines, which maximizes the rate of change of magnetic flux through the coil, thus maximizing the emf.

– However, if we consider the geometry and orientation of the coil, the emf produced actually becomes zero when the coil is exactly perpendicular to the magnetic field because there is no change in the magnetic flux linked with it at that instant.

– Therefore, the assertion is false, even though the reason correctly describes the situation for maximum change in flux.

Key Concept: Faraday’s Law

a) -5 V
[Solution Description] According to Faraday’s Law, the induced emf $$(\epsilon)$$ is given by the rate of change of magnetic flux. The formula is:

$$\epsilon = -\frac{\Delta \Phi}{\Delta t}$$

where $$\Delta \Phi = \Phi_f – \Phi_i = 15 – 5 = 10 \, \text{Wb},$$ and $$\Delta t = 2 \, \text{s}.$$ Thus,

$$\epsilon = -\frac{10}{2} = -5 \, \text{V}$$
The negative sign indicates the direction of emf opposes the change in flux.

Key Concept: Area Vector Concept

d) Assertion is false, but Reason is true.

[Solution Description]

The assertion states that the area vector is not important when calculating magnetic flux, which contradicts the definition of magnetic flux itself. Magnetic flux is calculated using the dot product of the magnetic field vector and the area vector. The reason provided correctly describes how the area vector comes into play in determining the magnetic flux. Therefore, the assertion is false, but the reason is true.

Key Concept: Induced EMF Calculation, Real-world Application

b) 2.0 V
[Solution Description] To determine the total induced emf, account for both motion-induced and field-change-induced emfs.

First, calculate the motion-induced emf using the formula:
$$
\epsilon_{\text{motion}} = B \cdot l \cdot v = 0.4 \, \text{T} \cdot 2 \, \text{m} \cdot 3 \, \text{m/s} = 2.4 \, \text{V}.
$$

Next, calculate the field-change-induced emf using the rate of change of the magnetic field:
$$
\epsilon_{\text{field}} = -l \cdot \frac{dB}{dt} = -2 \, \text{m} \cdot 0.2 \, \text{T/s} = -0.4 \, \text{V}.
$$

Combine both components to find the total induced emf:
$$
\epsilon_{\text{total}} = \epsilon_{\text{motion}} + \epsilon_{\text{field}} = 2.4 \, \text{V} – 0.4 \, \text{V} = 2.0 \, \text{V}.
$$

Hence, the total induced emf is $$2.0 \, \text{V}$$.

Key Concept: Advanced Conceptual Understanding, Real-world Application

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

[Solution Description]

The assertion states that self-inductance plays a vital role in controlling the timing in electronic circuits. This is true because self-inductance influences how quickly current can change through the coil, affecting the time-dependent behavior of the circuit.

The reason explains that self-inductance creates a phase difference between voltage and current, impacting the circuit’s resonance frequency. This is also true as self-inductance contributes to the impedance in AC circuits, leading to a phase shift.

However, the reason provided does not directly explain why self-inductance is crucial for controlling timing specifically; rather, it describes an outcome or effect of self-inductance. Controlling timing involves many factors including the overall circuit design, where inductance is just one part of the consideration but not solely defined by phase difference.

Therefore, both statements are true, but the Reason does NOT correctly explain the Assertion, leading us to select option (b).

Key Concept: Complex Problem Solving, Advanced Reasoning

d) Assertion is false, but Reason is true.

[Solution Description]

To solve this problem, we first need to understand how the position of the loop affects the magnetic flux and consequently the induced emf.

The magnetic flux $\Phi$ through the loop is given by the equation:

$$\Phi = BA \cos \theta$$

where B is the magnetic field, A is the area of the loop, and $\theta$ is the angle between the normal to the loop and the magnetic field.

According to Faraday’s Law, the induced emf $(\epsilon)$ is related to the rate of change of magnetic flux:

$$\epsilon = -\frac{d\Phi}{dt}$$

(A) states that the induced emf is maximum when the loop is perpendicular to the magnetic field. When the loop is perpendicular, $\theta = 90^\circ,$ thus $\cos \theta = 0.$ However, for maximum emf, it should be when $$\theta = 0^\circ$$ or $$180^\circ$$, making $$\cos \theta = 1$$ or -1, maximizing the magnitude of $\Phi,$ so (A) is false.

For (R), when the plane of the loop is parallel to the magnetic field, the angle $\theta$ would actually make the cosine function maximum, but if the loop moves such that $\theta$ remains constant while being parallel, there’s no change in magnetic flux. So the statement about the rate of change becoming zero is true because moving parallel doesn’t alter flux magnitude over time without changes in A or B.

Therefore, Assertion (A) is false, but Reason (R) is true.

Key Concept: Design Consideration, Real-World Application

c) Ensuring effective cooling and heat dissipation
[Solution Description]
Designing an AC generator for a nuclear power plant involves several critical considerations to optimize performance and safety. One major design challenge is ensuring a stable output, which requires minimizing mechanical vibrations and ensuring efficient heat dissipation due to large-scale operations.

Among the listed options, maintaining structural integrity under high thermal stresses and optimizing cooling systems are paramount because they prevent overheating and potential failures. These factors directly impact efficiency and safety.

Key Concept: Material Understanding

c) Assertion is true, but Reason is false.

[Solution Description] The assertion that brushes in AC generators are made of carbon is true, as carbon brushes provide good conductivity and wear resistance. However, the reason given is false because carbon is actually used for its conductive properties, not for insulation.

Key Concept: Relative Motion

b) The relative motion between the magnet and the coil
[Solution Description] According to Faraday’s law of electromagnetic induction, an emf is induced in a coil if there is a change in magnetic flux through the coil due to relative motion between the coil and the magnet. The movement of the magnet toward the coil changes the magnetic flux linked with the coil, resulting in the induction of emf.

Key Concept: Direct Relationship

c) Current grows slower
[Solution Description] In an inductive circuit, increasing resistance leads to slower growth of current. This is because higher resistance reduces the amount of current flowing through the circuit for the same induced emf, thus slowing down the rate at which current reaches its maximum value.

Key Concept: Basic Definition

c) The interaction between magnetic fields of solenoids S1 and S2
[Solution Description]
Mutual inductance $$M_{12}$$ is defined as the measurement of the electromotive force induced in one solenoid (S2) when the current through another solenoid (S1) changes. It basically quantifies the coupling between two circuits or solenoids.

Key Concept: Method Comparison

c) Utilizing Faraday’s Law of Induction
[Solution Description]
In addition to using the linear motion formula $$ \epsilon = Blv $$, we can use Faraday’s law of electromagnetic induction, which states that the emf induced is equal to the rate of change of magnetic flux through the loop. This approach involves calculating the changing area or orientation of the loop in relation to the magnetic field.

Key Concept: Direction of Force

b) Perpendicular to v and B

[Solution Description] The direction of the Lorentz force acting on a charge in a moving conductor within a magnetic field is given by the right-hand rule. The force is perpendicular to both the velocity vector v and the magnetic field B. Thus, the correct answer is that it is perpendicular to both these vectors.

Key Concept: Faraday’s Insight

d) Induced emf is proportional to the rate of change of magnetic flux.
[Solution Description] Faraday discovered that the electromotive force (emf) induced in a circuit is directly proportional to the rate of change of the magnetic flux through the circuit. This relationship can be mathematically expressed as $$ \epsilon = -\frac{d\Phi_B}{dt} $$, where $$ \epsilon $$ is the induced emf and $$ \frac{d\Phi_B}{dt} $$ is the rate of change of magnetic flux.

Key Concept: Role of Slip Rings

b) To maintain electrical contact with the rotating coil
[Solution Description]
Slip rings are used in an AC generator to maintain electrical contact between the rotating coil (armature) and the external circuit. They allow the coil to rotate freely while providing a continuous path for current to flow without twisting wires.

Key Concept: Historical Context

d) Joseph Henry
[Solution Description] The unit ‘Henry’ is named in honor of the American scientist Joseph Henry who made significant contributions to the field of electromagnetism.

Key Concept: Calculation Based

c) $$2.25 \, J$$
[Solution Description] The energy stored in an inductor is given by the formula $$W = \frac{1}{2} L I^2,$$ where L is the self-inductance and I is the current.

Using the given values $$L = 0.5 \, H and I = 3 \, A:$$
$$W = \frac{1}{2} \times 0.5 \times (3)^2$$

Calculating further:
$$W = \frac{1}{2} \times 0.5 \times 9$$
$$W = 0.25 \times 9 = 2.25 \, J$$

Therefore, the energy stored is $$2.25 \, J.$$

Key Concept: Motional EMF

b) $$Blv$$
[Solution Description] The motional emf induced when a conductor of length l moves with velocity v perpendicular to a magnetic field B is given by the formula $$\text{emf} = Blv$$.

Key Concept: Faraday’s Law Application

c) They are induced by a changing magnetic field.
[Solution Description] According to Faraday’s law of electromagnetic induction, a change in magnetic flux through a closed loop induces an electromotive force (EMF) in the loop. When a conductor moves relative to a magnetic field, or when the magnetic field around a stationary conductor changes, this induces loops of current within the conductor known as eddy currents.

Key Concept: Uniform Field Calculation

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

[Solution Description]

Assertion states that the magnetic flux is zero when the plane of the loop is perpendicular to the magnetic field. According to the formula for magnetic flux, $$F_B = B \cdot A = BA \cos \theta$$, if $$\theta = 90^\circ$$, then $$\cos 90^\circ = 0$$. This means the magnetic flux will indeed be zero. Therefore, the reason given is correct because it accurately describes the situation where the angle between the area vector and the magnetic field is 90 degrees.

Key Concept: Units of Inductance

c) Henry
[Solution Description] The SI unit of mutual inductance is Henry, denoted by H.

Key Concept: Direction of Current

d) It opposes the change in magnetic flux
[Solution Description] Lenz’s Law states that the direction of the induced current in a coil will be such that it opposes the change in magnetic flux that produced it. This ensures conservation of energy by opposing the cause of induction.