Class 12 Physics Chapter 4 Moving Charges and Magnetism

Key Concept: Current Density Concept

d) The drift velocity is tripled.

[Solution Description]

According to the relationship between current density (J), current (I), and drift velocity $(v_d)$, we have:

$J = n e v_d$

where n is the number of charge carriers per unit volume and e is the charge of an electron.

With J being tripled, the relationship becomes:

$3J = n e v’_d$

This implies:

$v’_d = 3v_d$

Thus, the new drift velocity is three times the original drift velocity.

Key Concept: Field Interaction, Theoretical Implications

c) Assertion is true, but Reason is false.

[Solution Description] To solve this assertion-reason question, we need to analyze both statements separately.

First, the assertion states that an external magnetic field can alter the internal field distribution in a toroidal system. This is true because the principle of superposition allows external fields to interact with internal fields, potentially leading to constructive or destructive interference effects.

Next, the reason claims that such superposition will always enhance field confinement and uniformity. This statement is false. While superposition can lead to various outcomes, assuming it always enhances confinement and uniformity without considering the exact nature and direction of fields involved is incorrect. Interference could distort the field instead.

Therefore, while both assertions about field interactions are theoretically plausible, the reason does not correctly explain the assertion’s potential outcomes, as it makes a generalized and inaccurate claim about superposition effects. Thus, the correct option is c).

Key Concept: Field at Center

b) $\frac{\mu_0 I}{2R}$
[Solution Description]
The magnetic field at the center of a circular loop is given by the formula:
$B = \frac{\mu_0 I}{2R}$
where $\mu_0$ is the permeability of free space.

Key Concept: Basic Concept

b) Ampere meter squared (A m²)
[Solution Description] Magnetic moment (m) has units of A m² or J/T as given in the syllabus.

Key Concept: Complex Motion Analysis, Helical Motion Dynamics

c) Assertion is true, but Reason is false.

[Solution Description]

The assertion states that when a charged particle moves in a magnetic field with velocity components both perpendicular and parallel to the field, it forms a helical path. The radius of the circular part of this path is given by $r = \frac{mv_{\perp}}{qB}$, which indicates that the radius is directly proportional to the perpendicular component $v_{\perp}$ of velocity and inversely proportional to the magnetic field strength $B$.

In contrast, the reason claims that the pitch of the helical path depends only on the component of velocity parallel to the magnetic field $v_{||}$. The pitch $p = \frac{2\pi mv_{||}}{qB}$ shows dependency on both the parallel velocity $v_{||}$ and the magnetic field strength $B$, contradicting the statement in the Reason.

Thus, while both the Assertion and Reason are true regarding their respective aspects of motion dynamics, the Reason incorrectly describes the full set of factors affecting the pitch as being independent of the magnetic field strength. Therefore, the correct option is (c) Assertion is true, but Reason is false.

Key Concept: Complex Geometry, Limitations of Law

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

[Solution Description] Ampère’s Law is given by the integral form $\oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{\text{enclosed}}$, which holds under conditions where symmetry allows the simplification of $\mathbf{B}$ along an Amperian loop. In cases like a non-uniform cross-sectional coil, symmetry is not present, making it difficult or impossible to determine a consistent magnetic field magnitude and direction that satisfies the path integral of the law. The lack of symmetry affects the ability to apply Ampère’s Law directly as the field is not predictable over complex geometries. Thus, while both the assertion and reason are true, the reason provides a valid explanation for why Ampère’s Law struggles in such configurations.

Key Concept: Basic Concept

d) A doughnut-shaped coil or ring
[Solution Description] A torus, in the context of magnetic fields, is essentially a doughnut-shaped object or structure through which current can flow, creating a magnetic field within its central cavity.

Key Concept: Magnetic Force Zero Condition

d) Both a) and c)

[Solution Description] The magnetic force on a charge becomes zero under two conditions: either when the charge is not moving (velocity v = 0), or when the velocity and magnetic field are parallel to each other. In both cases, the cross product in the Lorentz force equation becomes zero, resulting in no magnetic force.

Key Concept: Helical Motion

b) Helical

[Solution Description]

When a charged particle has a velocity component along with the direction of the magnetic field and another perpendicular to it, the particle follows a helical path. The perpendicular component causes circular motion, while the parallel component advances the motion along the axis of the magnetic field.

Key Concept: Conservation of Momentum, Electromagnetic Field Consideration

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

[Solution Description] In systems where currents vary with time, electrical and magnetic fields are dynamically changing, leading to a situation where the sum of mechanical momenta of the interacting conductors may not be conserved as it would in static or steady situations. This is because the electromagnetic fields themselves carry momentum, which must be considered to maintain an overall conservation law. Thus, while the assertion corrects the assumption that only mechanical forces contribute to momentum conservation, the reason provides a valid explanation by indicating the role of electromagnetic field momentum.

Key Concept: Unit Identification

d) Tesla
[Solution Description] The SI unit of magnetic field strength is Tesla (T), which measures the magnitude of the magnetic field.

Key Concept: Basic Definition

c) Principle of superposition

[Solution Description] The magnetic field follows the principle of superposition, meaning that the combined magnetic field from several sources is equal to the vector addition of the individual fields.

Key Concept: Conceptual Understanding

c) The magnetic field reverses its direction.

[Solution Description] When the direction of the current in a wire is reversed, the direction of the magnetic field around the wire also reverses. According to the right-hand rule, if you point your thumb in the direction of the current, your fingers curl in the direction of the magnetic field lines. Reversing the current means flipping the direction of both thumb and fingers, thus reversing the magnetic field.

Key Concept: Field Calculation

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

[Solution Description]

To calculate the magnetic field inside a solenoid, Ampere’s Law is applied which provides us with the formula:

$B = \mu_0 n I$

where B is the magnetic field, $\mu_0$ is the permeability of free space, n is the number of turns per unit length, and I is the current flowing through the solenoid.

This formula effectively captures how both the density of wire turns (n) and the current (I) contribute to generating the magnetic field inside the solenoid. Therefore, both the Assertion and Reason are true, and the Reason correctly explains the Assertion as the parameters mentioned in the Reason are components of the given formula.

Key Concept: Formula Application

c) $dB = \frac{\mu_0}{4\pi} \frac{I \, dl \times r}{r^3}$
[Solution Description] According to the Biot-Savart Law, the magnetic field $dB$ produced by a small current element $dl$ is given by the formula
$dB = \frac{\mu_0}{4\pi} \frac{I \, dl \times r}{r^3}.$
This shows the magnetic field as proportional to the current $I$, length $|dl|$ and inversely proportional to the cube of distance $r$. The direction is perpendicular to the plane containing $dl$ and $r$.

Key Concept: Frequency Understanding

b) $\nu = \frac{q B}{2 \pi m}$
[Solution Description]
The cyclotron frequency $\nu$ of a charged particle with charge q, magnetic field strength B, and mass m is independent of the particle’s velocity or energy and is given by

$\nu = \frac{q B}{2 \pi m}$

This formula shows how the frequency depends on the charge, magnetic field, and mass.

Key Concept: Field Calculation

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

[Solution Description] To solve this problem, we need to understand that for a long straight current-carrying wire, Ampère’s Circuital Law simplifies the calculation of the magnetic field due to symmetry. The formula $B = \frac{\mu_0 I}{2\pi r}$ can be derived as follows:

1. Choose a circular Amperian loop of radius r centered on the wire. By symmetry, the magnetic field must be constant in magnitude and directed tangentially around any circle surrounding the wire.

2. Calculate the line integral of the magnetic field along this path:

$\oint B \cdot dl = B(2\pi r)$

where B is constant along the path, and dl represents an infinitesimal segment of the loop.

3. According to Ampère’s Circuital Law:

$\oint B \cdot dl = \mu_0 I_e$

4. Equating both expressions gives:

$B(2\pi r) = \mu_0 I$

5. Solving for B, we have:

$B = \frac{\mu_0 I}{2\pi r}$

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

Key Concept: Basic Formula

c) $\tau = IAB \sin \theta$
[Solution Description] The formula for the torque $(\tau)$ on a rectangular current loop with current I, area A, and angle $\theta$ between the normal to the loop’s plane and the magnetic field B is given by:
$\tau = IAB \sin \theta$

Key Concept: Angle Influence

d) The magnetic field is zero
[Solution Description] When the angle $\theta$ is 0 degrees, dl is in the same direction as r, making the cross product $dl \times r$ equal to zero. Therefore, the magnetic field is zero.

Key Concept: Historical Insight

d) Hans Christian Oersted
[Solution Description] The discovery of the relationship between electricity and magnetism was made by Hans Christian Oersted. He observed that a current-carrying wire causes a deflection in a nearby magnetic compass needle, establishing a fundamental connection between the two phenomena.

Key Concept: Basic Concept

c) A long wire wound in a helical shape with closely spaced turns
[Solution Description] A solenoid is defined as a long wire wound in the form of a helix with closely spaced turns. This structure allows it to generate a uniform magnetic field when current passes through it.

Key Concept: Design Optimization, Conversion Challenges

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

[Solution Description]

Increasing the number of turns $N$ in the coil of a galvanometer increases the torque for a given current due to the formula $\tau = NIAB$, thereby increasing its sensitivity per unit current when considering the equation for current sensitivity $\phi/I = NAB/k$. However, while this is true for general sensitivity, in the specific case of conversion into an ammeter where accuracy is paramount, the added resistance from extra turns could indeed impact the device’s ability to measure high currents accurately since it might necessitate adjusting shunt resistance to maintain a low equivalent resistance. Therefore, both Assertion and Reason are correct, but the increase in resistance discussed in the Reason does not directly explain how sensitivity is enhanced by the addition of turns.

Key Concept: Force on Current-Carrying Rod

c) 1.8 N

[Solution Description] The force on a current-carrying conductor can be calculated using $F = I l B$. Here, $I = 4 \, \text{A}$, $l = 1.5 \, \text{m}$, and $B = 0.3 \, \text{T}$. Substituting these values:

$F = 4 \times 1.5 \times 0.3 = 1.8 \, \text{N}$

Therefore, the force is $1.8 \, \text{N}$.

Key Concept: Right-Hand Rule

b) Downwards

[Solution Description] According to the right-hand thumb rule, if you curl the fingers of your right hand in the direction of current flow through the loop (clockwise when viewed from above), your thumb points in the direction of the axial magnetic field, which is downwards below the loop.

Key Concept: Basic Concept

d) Tesla (T)
[Solution Description] The unit of magnetic field strength B is Tesla (T). This is a fundamental concept related to the measurement of magnetic fields.

Key Concept: Formula Application

b) $5 \times 10^{-5}\, \text{N/m}$
[Solution Description] The formula for the force per unit length between two parallel conductors is given by:
$f_{ba} = \frac{\mu_0 I_a I_b}{2\pi d}$
Substituting the values:
$I_a = 5\, \text{A}$, $I_b = 10\, \text{A}$, $d = 0.2\, \text{m}$, we get:
$f_{ba} = \frac{(4\pi \times 10^{-7}) \times 5 \times 10}{2\pi \times 0.2}$
$= \frac{20 \times 10^{-7}}{0.2}$
$= 10^{-5} \times 100$
$= 10^{-3}\, \text{N/m}$
Simplifying gives us $f_{ba} = 5 \times 10^{-5}\, \text{N/m}$.

Key Concept: Field and Torque Relationship, Solenoid Field Analysis

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

[Solution Description]

To analyze this, let’s consider the relationship between the torque $\tau$ on a current loop in a magnetic field B. The expression for torque is given by:

$\tau = mB\sin(\theta)$

where m is the magnetic dipole moment, B is the magnetic field strength, and $\theta$ is the angle between the normal to the plane of the loop and the magnetic field.

In the case provided, when the plane of the current loop is parallel to the magnetic field inside a long solenoid, $\theta = 0^\circ$, making $\sin(0^\circ) = 0$. Thus, $\tau = mB \cdot 0 = 0$, which makes the assertion true.

For the reason, inside a long solenoid, the field lines are indeed uniform and run parallel to the axis of the solenoid due to the nature of the solenoid configuration. This provides a consistent direction for any inserted dipoles or loops with respect to the solenoid’s axis, confirming that the magnetic field is uniform along the axial direction.

Both the assertion and reason are true statements. However, the reason does not directly explain why the torque becomes zero; it merely describes the uniformity and direction of the field inside the solenoid.

Key Concept: Right-Hand Rule

c) Right-hand thumb rule

[Solution Description] The right-hand thumb rule is used to determine the direction of the magnetic field around a current-carrying conductor. According to this rule, if you hold the wire with your right hand such that the thumb points in the direction of the current, the curled fingers indicate the direction of the magnetic field.