Chapter 4: Moving Charges and Magnetism Long Questions | physics Class 12 CBSE

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Chapter 4: Moving Charges and Magnetism Long Questions | physics Class 12 CBSE | ByteNotes.in

Long Answer

Hard difficulty • Structured explanation

Hard

Question 1

Long Form

Derive the expression for the magnetic field at an axial point of a circular current loop and reduce it to the field at the centre. Also state the right-hand thumb rule for the field direction.

Consider a circular loop of radius R carrying current I, centred at origin in the y-z plane. For an element dl at the loop, the displacement to axial point P at distance x is r = (x² + R²)^(1/2). Since dl is perpendicular to r, |dl × r| = r dl and by Biot-Savart: dB = μ₀I dl / [4π(x² + R²)].
The perpendicular components (dB⊥) from diametrically opposite elements cancel by symmetry; only the axial components dBx = dB cosθ survive, where cosθ = R/(x² + R²)^(1/2).
Integrating over the full loop (∮dl = 2πR): B = μ₀IR²/[2(x² + R²)^(3/2)], directed along the positive x-axis (axis of the loop).
At the centre, x = 0: B₀ = μ₀I/(2R). The field at the centre is stronger for a smaller loop radius.
Right-hand thumb rule: curl the fingers of the right hand around the loop in the direction of current flow; the extended thumb points in the direction of the magnetic field.

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Long Answer

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Hard

Question 2

Long Form

Using Ampere's circuital law, derive the expression for the magnetic field inside and outside a long straight wire of circular cross-section carrying a uniformly distributed steady current I.

For r > a (outside): Choose a circular amperian loop of radius r concentric with wire. By symmetry, B is tangential and uniform along the loop. Ampere's law gives B(2πr) = μ₀I, so B = μ₀I/(2πr). Field decreases as 1/r outside the wire.
For r < a (inside): The enclosed current Ie = I(πr²/πa²) = Ir²/a², since current is uniformly distributed. Applying Ampere's law: B(2πr) = μ₀Ir²/a², giving B = (μ₀I/2πa²)r. Field increases linearly with r inside the wire.
At the surface (r = a): both expressions give B = μ₀I/(2πa), confirming continuity of the field at the boundary.
The field is maximum at the surface r = a: Bmax = μ₀I/(2πa); inside it grows linearly (B ∝ r) and outside it falls as 1/r.
This example has cylindrical symmetry, which is why Ampere's law can be applied directly. Ampere's circuital law holds for steady currents and is analogous to Gauss's law in electrostatics.

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Long Answer

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Hard

Question 3

Long Form

Derive the expression for the magnetic field inside a long solenoid using Ampere's circuital law and discuss the nature of the field.

A long solenoid has n turns per unit length, length >> radius, enamelled wire to insulate turns. The field inside is uniform and along the axis; the field outside approaches zero.
Choose a rectangular Amperian loop abcd: side ab (length h) lies inside, parallel to axis; side cd lies outside. Contributions from transverse sides bc and ad are zero (B perpendicular to them).
Contribution from cd is zero (B ≈ 0 outside). Only side ab contributes: ∮B·dl = Bh. Total enclosed current: Ie = I(nh), where nh is the number of turns in length h.
Applying Ampere's law: Bh = μ₀I(nh), giving B = μ₀nI. This is uniform inside regardless of the position along the axis.
The solenoid is used to produce a uniform magnetic field. A large field can be achieved by inserting a soft iron core inside the solenoid. The field direction is given by the right-hand rule.

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Long Answer

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Medium

Question 4

Long Form

Explain the principle, construction, and working of a moving coil galvanometer. Derive the expression for its deflection.

Principle: A current-carrying coil placed in a magnetic field experiences a torque τ = NIAB, which causes angular deflection proportional to the current.
Construction: It consists of a rectangular coil of N turns wound on a non-magnetic frame, suspended on a pivot in a uniform radial magnetic field created by concave pole pieces and a cylindrical soft iron core. A spring provides a restoring torque.
Working: When current I flows, the magnetic torque NIAB rotates the coil; the spring provides counter-torque kφ. At equilibrium: kφ = NIAB.
Deflection expression: φ = (NAB/k)I. Since NAB/k is constant for a given galvanometer, deflection is directly proportional to current (linear scale).
The radial field ensures sinθ = 1 at all deflections, maintaining linearity. The soft iron core concentrates the field and makes it radial. Current sensitivity = NAB/k; voltage sensitivity = NAB/(kR).

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Long Answer

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Hard

Question 5

Long Form

Derive the force per unit length between two long parallel current-carrying conductors and state the rule for direction of force. How is this used to define the ampere?

Consider conductors a and b separated by d, carrying currents Ia and Ib. Conductor a produces field Ba = μ₀Ia/(2πd) at the location of conductor b (by Ampere's law).
Conductor b of length L in field Ba experiences force Fba = IbLBa = μ₀IaIbL/(2πd). By Newton's third law, Fab = –Fba (equal and opposite).
Force per unit length: fba = μ₀IaIb/(2πd). Parallel currents attract; antiparallel currents repel. This is opposite to the behavior of electric charges.
Definition of ampere: One ampere is the current in each of two very long, straight, parallel conductors of negligible cross-section, placed 1 m apart in vacuum, that produces a force of 2 × 10⁻⁷ N/m on each conductor.
Substituting Ia = Ib = 1 A, d = 1 m in fba = μ₀IaIb/(2πd): f = (4π × 10⁻⁷ × 1 × 1)/(2π × 1) = 2 × 10⁻⁷ N/m, consistent with the definition.

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Question 6

Long Form

Derive the expression for the torque on a rectangular current loop in a uniform magnetic field when the loop's plane makes an arbitrary angle with the field. Express the result in vector form.

Let the rectangular loop of dimensions a × b carry current I in uniform field B. Let θ be the angle between the area vector A (normal to loop) and B.
Forces on arms BC and DA are equal and opposite along the coil axis — they cancel giving no torque. Forces on arms AB and CD have magnitude F1 = F2 = IbB each.
These forces form a couple with perpendicular distance a sinθ, giving torque: τ = F1 × (a sinθ) = IbB × a sinθ = IAB sinθ, where A = ab.
For N turns: τ = NIAB sinθ. Defining magnetic moment m = NIA, the torque in vector form is τ = m × B.
When θ = 90° (plane parallel to B): τ = NIAB (maximum). When θ = 0° (m ∥ B): τ = 0 (stable equilibrium). This result applies to any planar loop, not just rectangular ones.

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Long Answer

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Medium

Question 7

Long Form

Analyse how a galvanometer is converted into an ammeter and a voltmeter. Compare the resistance requirements for each and explain why an ideal ammeter has zero resistance while an ideal voltmeter has infinite resistance.

Ammeter conversion: A small shunt resistance rs is connected in parallel with the galvanometer coil. Most current passes through rs; only a small fraction goes through the galvanometer. Effective resistance ≈ RGrs/(RG + rs) ≈ rs (since RG >> rs).
Voltmeter conversion: A large resistance R is connected in series with the galvanometer. The voltmeter is connected in parallel with the section of the circuit being measured and must draw negligible current. Effective resistance = RG + R ≈ R.
An ideal ammeter has zero resistance: since it is connected in series, any resistance it adds reduces the current being measured. Zero resistance ensures no disturbance to the circuit.
An ideal voltmeter has infinite resistance: since it is connected in parallel, it must draw zero current to avoid altering the voltage distribution. Infinite resistance ensures no current is diverted.
Real ammeters and voltmeters are calibrated after conversion. The shunt value determines the full-scale current range; the series resistance determines the full-scale voltage range.

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Long Answer

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Hard

Question 8

Long Form

Discuss the motion of a charged particle in a uniform magnetic field. Derive expressions for the radius and period of circular motion and explain what happens when the velocity has a component along the field.

A charge q moving with speed v perpendicular to B experiences magnetic force qvB (Lorentz force), which is perpendicular to v. This acts as centripetal force: qvB = mv²/r.
Solving: radius r = mv/(qB). A larger momentum gives a larger radius. Period T = 2πr/v = 2πm/(qB), and cyclotron frequency ν = 1/T = qB/(2πm) — both independent of speed.
The magnetic force does no work (always perpendicular to v), so speed remains constant; only the direction changes, resulting in uniform circular motion in the plane perpendicular to B.
If v has a component v‖ parallel to B: this component is unaffected (magnetic force on v‖ is zero). The perpendicular component v⊥ produces circular motion. The combined effect is helical motion.
Pitch of the helix: p = v‖ × T = 2πmv‖/(qB). Radius of helix = mv⊥/(qB). The independence of ν from energy is the key principle behind the cyclotron accelerator.

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Long Answer

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Hard

Question 9

Long Form

Compare the Biot-Savart law with Ampere's circuital law — their physical content, applicability, and relationship to the corresponding laws in electrostatics.

Biot-Savart law gives the field dB due to an infinitesimal current element I dl at a point P: dB = (μ₀/4π)(I dl × r)/r³. It applies to arbitrary current configurations by integration. It is analogous to Coulomb's law (electric field due to a point charge).
Ampere's circuital law states ∮B·dl = μ₀Ie and relates the line integral of B around a closed loop to the enclosed current. It is analogous to Gauss's law (flux of E through a closed surface relates to enclosed charge).
Ampere's law is derived from the Biot-Savart law — it is not an independent law. Similarly, Gauss's law can be derived from Coulomb's law. Both Ampere and Gauss express the same physical content in integral form over boundaries.
Ampere's law is most useful for symmetric configurations (infinite wires, solenoids, toroids) where an appropriate amperian loop can be chosen to simplify the calculation. Biot-Savart is more general but requires integration.
Both laws hold for steady currents in the chapter's context. Ampere's law in its full form (with Maxwell's displacement current) is needed for time-varying fields, a topic treated in Chapter 8.

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Long Answer

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Hard

Question 10

Long Form

Analyse the circular current loop as a magnetic dipole. Show the analogy with an electric dipole and explain the fundamental difference between the two.

For a circular loop of radius R carrying current I, the axial field at distance x >> R simplifies to B ≈ μ₀IR²/(2x³) = μ₀(IA)/(2πx³) × π = μ₀(2m)/(4πx³), where m = IA is the magnetic moment.
This is analogous to the axial electric field of an electric dipole: E = 2pe/(4πε₀x³). The analogy maps m → pe, B → E, μ₀ → 1/ε₀.
Similarly, the field in the equatorial plane (plane of loop, x >> R) is B ≈ μ₀m/(4πx³), analogous to the equatorial field E ≈ pe/(4πε₀x³) of an electric dipole.
Fundamental difference: An electric dipole consists of two electric monopoles (charges +q and -q). A magnetic dipole (current loop) is the most elementary magnetic element — magnetic monopoles do not exist.
The torque on both dipoles in their respective fields follows the same form: τ = m × B (magnetic) and τ = pe × E (electric). This analogy is powerful for understanding the behavior of atomic magnetic moments.

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Chapter 4: Moving Charges and Magnetism | Long Questions

Derive the expression for the magnetic field at an axial point of a circular current loop and reduce it to the field at the centre. Also state the right-hand thumb rule for the field direction.

Using Ampere's circuital law, derive the expression for the magnetic field inside and outside a long straight wire of circular cross-section carrying a uniformly distributed steady current I.

Derive the expression for the magnetic field inside a long solenoid using Ampere's circuital law and discuss the nature of the field.

Explain the principle, construction, and working of a moving coil galvanometer. Derive the expression for its deflection.

Derive the force per unit length between two long parallel current-carrying conductors and state the rule for direction of force. How is this used to define the ampere?

Derive the expression for the torque on a rectangular current loop in a uniform magnetic field when the loop's plane makes an arbitrary angle with the field. Express the result in vector form.

Analyse how a galvanometer is converted into an ammeter and a voltmeter. Compare the resistance requirements for each and explain why an ideal ammeter has zero resistance while an ideal voltmeter has infinite resistance.

Discuss the motion of a charged particle in a uniform magnetic field. Derive expressions for the radius and period of circular motion and explain what happens when the velocity has a component along the field.

Compare the Biot-Savart law with Ampere's circuital law — their physical content, applicability, and relationship to the corresponding laws in electrostatics.

Analyse the circular current loop as a magnetic dipole. Show the analogy with an electric dipole and explain the fundamental difference between the two.

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