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Class 12 Physics Notes Chapter 4 (Moving charges and magnetism) – Physics Part-I Book

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Philoid

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Physics Part-I
Detailed Notes with MCQs of Chapter 4: Moving Charges and Magnetism from your NCERT Physics Part-I textbook. This is a crucial chapter, bridging electricity and magnetism, and forms the basis for many questions in competitive government exams. Pay close attention to the concepts, formulas, and direction rules.

Chapter 4: Moving Charges and Magnetism - Detailed Notes

1. Introduction & Oersted's Discovery:

  • Hans Christian Oersted (1820) discovered that an electric current flowing through a wire deflects a nearby magnetic compass needle.
  • This established the fundamental link: Moving charges (currents) produce magnetic fields.
  • Conversely, magnetism can also affect moving charges.

2. Magnetic Force (Lorentz Force):

  • Force on a Moving Charge: A charge 'q' moving with velocity 'v' in a magnetic field 'B' experiences a magnetic force 'Fm'.
    • Formula: Fm = q (v × B)
    • Magnitude: Fm = |q| v B sinθ, where θ is the angle between v and B.
    • Direction: Given by the Right-Hand Palm Rule (or Fleming's Left-Hand Rule).
      • Right-Hand Palm Rule: Point fingers in the direction of B, thumb in the direction of v (for +q). The force Fm acts outwards from the palm. For a negative charge (-q), the force is in the opposite direction.
      • Fleming's Left-Hand Rule: Stretch thumb, forefinger, and middle finger mutually perpendicular. If Forefinger points to Field (B) and Middle finger points to Current (or direction of v for +q), then Thumb points to the direction of Force (Fm).
    • Key Points:
      • Force is zero if the charge is stationary (v=0).
      • Force is zero if the charge moves parallel or anti-parallel to the magnetic field (θ = 0° or 180°).
      • Force is maximum when the charge moves perpendicular to the magnetic field (θ = 90°), Fm = |q| v B.
      • The magnetic force is always perpendicular to both v and B. Therefore, the magnetic force does no work on the charge (W = F ⋅ d = 0, as F is perpendicular to displacement d which is along v).
      • Since no work is done, the magnetic force cannot change the kinetic energy or speed of the charged particle, it only changes its direction of motion.
  • Unit of Magnetic Field (B):
    • SI Unit: Tesla (T). 1 T = 1 N s / (C m) = 1 N / (A m)
    • Other Unit: Gauss (G). 1 G = 10⁻⁴ T. (Earth's magnetic field is ~ 0.5 G)
  • Lorentz Force (Combined Electric and Magnetic Fields): If a charge 'q' moves in a region with both electric field 'E' and magnetic field 'B', the total force is the vector sum of electric force (qE) and magnetic force (q(v × B)).
    • Formula: F = qE + q(v × B) = q [E + (v × B)]

3. Motion of a Charged Particle in a Uniform Magnetic Field:

  • Case 1: v perpendicular to B:
    • The magnetic force (qvB) acts perpendicular to v, providing the necessary centripetal force for circular motion.
    • Path: Circular
    • Centripetal Force = Magnetic Force: mv²/r = qvB
    • Radius of circular path: r = mv / qB (r ∝ momentum, r ∝ 1/B)
    • Angular frequency (Cyclotron frequency): ω = v/r = qB / m (Independent of v and r)
    • Time period: T = 2π/ω = 2πm / qB (Independent of v and r)
    • Frequency: f = 1/T = qB / 2πm (Independent of v and r)
  • Case 2: v has components parallel (v∥) and perpendicular (v⊥) to B:
    • The perpendicular component (v⊥) causes circular motion (radius r = mv⊥ / qB).
    • The parallel component (v∥) remains unchanged (as force due to it is zero).
    • Resultant Path: Helical
    • Pitch of the helix (distance moved along B in one revolution): p = v∥ × T = v∥ (2πm / qB)

4. Velocity Selector:

  • A region with crossed (perpendicular) uniform electric (E) and magnetic (B) fields.
  • By adjusting E and B, only charges with a specific velocity 'v' pass undeflected.
  • Condition for undeflected motion: Electric Force = Magnetic Force => qE = qvB => v = E / B
  • Used in mass spectrometers and particle accelerators.

5. Cyclotron:

  • Device used to accelerate charged particles (like protons, deuterons, alpha particles) to high energies.
  • Principle: Uses both electric field (to accelerate) and magnetic field (to make particles move in a circular path and return to the accelerating region). The frequency of revolution of the charged particle in the magnetic field is independent of its speed and radius.
  • Working: Particles move in semi-circular paths inside two D-shaped hollow metal chambers ('Dees'). A high-frequency alternating electric field is applied across the gap between the Dees, accelerating the particles each time they cross the gap. A strong perpendicular magnetic field keeps them moving in circular paths of increasing radii.
  • Resonance Condition: The frequency of the applied AC voltage (oscillator frequency) must be equal to the cyclotron frequency: f_osc = f_c = qB / 2πm.
  • Maximum Kinetic Energy: K_max = (q² B² R²) / 2m, where R is the radius of the Dees.
  • Limitations: Cannot accelerate electrons (relativistic mass increase affects frequency) or neutral particles.

6. Biot-Savart Law:

  • Gives the magnetic field (dB) produced at a point due to a small current element (I dl). Analogous to Coulomb's law in electrostatics.
  • Formula (Vector Form): dB = (μ₀ / 4π) * (I dl × r) / r³
    • μ₀ = Permeability of free space = 4π × 10⁻⁷ T m / A
    • dl = length vector of the current element (direction of current)
    • r = position vector from the current element to the point where B is calculated.
    • r = |r|
  • Magnitude: dB = (μ₀ / 4π) * (I dl sinθ) / r², where θ is the angle between dl and r.
  • Direction: Given by the Right-Hand Thumb Rule: If you hold the wire with your right hand such that the thumb points in the direction of the current, the direction in which your fingers curl gives the direction of the magnetic field lines.

7. Applications of Biot-Savart Law:

  • Magnetic Field due to a Straight Current-Carrying Conductor:
    • Finite Length: B = (μ₀I / 4πa) * (sinφ₁ + sinφ₂), where 'a' is the perpendicular distance from the point to the wire, φ₁ and φ₂ are angles subtended by the ends of the wire at the point.
    • Infinite Length: B = μ₀I / 2πa (Field lines are concentric circles around the wire)
  • Magnetic Field at the Center of a Circular Current Loop:
    • B = μ₀I / 2R, where R is the radius of the loop.
  • Magnetic Field on the Axis of a Circular Current Loop:
    • B = (μ₀ N I R²) / [2 (R² + x²)^(3/2)], where N is the number of turns, R is the radius, I is the current, and x is the distance of the point from the center along the axis. (At the center, x=0, B = μ₀NI / 2R).

8. Ampere's Circuital Law:

  • Relates the line integral of the magnetic field around any closed loop (Amperian loop) to the net current passing through the area enclosed by the loop. Analogous to Gauss's law in electrostatics.
  • Statement: The line integral of the magnetic field B around any closed path is equal to μ₀ times the total current I_enclosed threading through the surface bounded by the closed path.
  • Formula: ∮ B ⋅ dl = μ₀ I_enclosed
  • Usefulness: Used to calculate magnetic fields in situations with high symmetry (infinite wire, solenoid, toroid).

9. Applications of Ampere's Circuital Law:

  • Magnetic Field due to an Infinitely Long Straight Wire: (Re-derivation)
    • Consider a circular Amperian loop of radius 'a' centered on the wire. B is tangential and constant in magnitude on the loop.
    • ∮ B dl cos(0°) = B ∮ dl = B (2πa)
    • B (2πa) = μ₀ I => B = μ₀I / 2πa
  • Magnetic Field Inside a Long Solenoid:
    • A solenoid is a long coil of wire wound in a tight helix.
    • Inside a long solenoid, the field is strong, uniform, and parallel to the axis. Outside, the field is nearly zero.
    • Formula: B = μ₀ n I
      • n = number of turns per unit length (N/L)
      • N = total number of turns, L = length of the solenoid.
  • Magnetic Field Inside a Toroid:
    • A toroid is a solenoid bent into the shape of a donut (hollow circular ring).
    • The magnetic field is confined within the toroidal core and is zero outside. Field lines are concentric circles.
    • Formula: B = μ₀ N I / (2πr) = μ₀ n I
      • N = total number of turns
      • r = radius of the Amperian loop within the toroid (average radius if thickness is small)
      • n = N / (2πr) = number of turns per unit length along the circumference.

10. Force Between Two Parallel Current-Carrying Wires:

  • Two parallel wires carrying currents I₁ and I₂ separated by a distance 'd'.
  • Wire 1 produces a magnetic field B₁ = μ₀I₁ / 2πd at the location of wire 2.
  • Wire 2 experiences a force due to this field B₁. Force on length L of wire 2: F₂ = I₂ L B₁ sin(90°) = I₂ L (μ₀I₁ / 2πd).
  • Force per unit length: F/L = μ₀ I₁ I₂ / 2πd
  • Nature of Force:
    • Currents in the same direction: Attractive force.
    • Currents in opposite directions: Repulsive force.
  • Definition of Ampere (SI Unit of Current): One Ampere is defined as that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed 1 meter apart in vacuum, would produce between these conductors a force equal to 2 × 10⁻⁷ Newton per meter of length.

11. Torque on a Current Loop in a Uniform Magnetic Field:

  • A rectangular loop (length l, breadth b, Area A=lb) carrying current 'I' placed in a uniform magnetic field 'B'. Let the normal to the loop make an angle θ with B.
  • Forces on sides parallel to B are zero or cancel out.
  • Forces on sides perpendicular to B (length l) are F = I l B, equal and opposite, forming a couple.
  • Torque (τ): τ = Force × perpendicular distance = (I l B) × (b sinθ) = I (lb) B sinθ = I A B sinθ
  • For a coil with N turns: τ = N I A B sinθ
  • Magnetic Dipole Moment (m): A current loop behaves like a magnetic dipole. Its magnetic dipole moment is defined as m = N I A, where A is the area vector (direction perpendicular to the loop, given by Right-Hand Thumb Rule - curl fingers along current, thumb gives direction of m and A).
  • Torque in Vector Form: τ = N I (A × B) = m × B (Analogous to τ = p × E in electrostatics)
  • Potential Energy of a magnetic dipole in B: U = - m ⋅ B = - m B cosθ

12. Moving Coil Galvanometer (MCG):

  • Principle: Torque acts on a current-carrying coil suspended in a uniform magnetic field (τ = N I A B sinθ).
  • Construction: Coil wound on a non-magnetic frame, suspended between pole pieces of a strong magnet (producing a radial magnetic field), pointer attached to the coil, spring providing restoring torque.
  • Radial Magnetic Field: Ensures that the plane of the coil is always parallel to the magnetic field lines (θ = 90°, sinθ = 1) for any orientation. This makes the deflecting torque directly proportional to the current (τ = NIAB).
  • Working: When current flows, deflecting torque τ_d = NIAB acts on the coil. The suspension wire/spring twists, producing a restoring torque τ_r = kφ, where k is the torsional constant and φ is the deflection angle.
  • In equilibrium: Deflecting Torque = Restoring Torque => NIAB = kφ
  • Deflection: φ = (NAB / k) I => φ ∝ I. The deflection is directly proportional to the current.
  • Sensitivity:
    • Current Sensitivity (IS): Deflection per unit current. IS = φ / I = NAB / k. (Units: rad/A or div/A)
    • Voltage Sensitivity (VS): Deflection per unit voltage. VS = φ / V = φ / (IR) = (NAB / k) * (1/R) = IS / R, where R is the galvanometer resistance. (Units: rad/V or div/V)
  • Conversion of Galvanometer:
    • Into Ammeter: To measure large currents. Connect a low resistance called shunt (S) in parallel with the galvanometer coil. S = (Ig G) / (I - Ig), where Ig is the full-scale deflection current, G is galvanometer resistance, I is the range of the ammeter. Effective resistance is very low (Rp = GS / (G+S)).
    • Into Voltmeter: To measure potential difference. Connect a high resistance (R) in series with the galvanometer coil. R = (V / Ig) - G, where V is the desired voltage range. Effective resistance is very high (Rs = G + R).

Multiple Choice Questions (MCQs):

  1. A proton enters a uniform magnetic field perpendicular to its direction of motion. Which of the following quantities will not change?
    (A) Velocity
    (B) Momentum
    (C) Kinetic Energy
    (D) Direction of motion

  2. The magnetic force on a current carrying wire of length L placed in a uniform magnetic field B is given by F = I L B sinθ. The direction of the force is given by:
    (A) Right-Hand Thumb Rule
    (B) Fleming's Left-Hand Rule
    (C) Fleming's Right-Hand Rule
    (D) Biot-Savart Law

  3. The SI unit of magnetic field strength is:
    (A) Weber (Wb)
    (B) Gauss (G)
    (C) Tesla (T)
    (D) Oersted (Oe)

  4. A long straight wire carries a current of 10 A. The magnetic field at a distance of 0.2 m from the wire is (μ₀ = 4π × 10⁻⁷ T m / A):
    (A) 10⁻⁵ T
    (B) 10⁻⁶ T
    (C) 2 × 10⁻⁵ T
    (D) 2 × 10⁻⁶ T

  5. Ampere's Circuital Law is analogous to which law in electrostatics?
    (A) Coulomb's Law
    (B) Ohm's Law
    (C) Kirchhoff's Law
    (D) Gauss's Law

  6. The magnetic field inside a long ideal solenoid is:
    (A) Directly proportional to the current (I)
    (B) Inversely proportional to the number of turns per unit length (n)
    (C) Zero
    (D) Inversely proportional to the current (I)

  7. Two parallel wires carry currents in the same direction. The force between them is:
    (A) Repulsive
    (B) Attractive
    (C) Zero
    (D) Depends on the magnitude of currents

  8. To convert a galvanometer into an ammeter, one needs to connect a:
    (A) High resistance in series
    (B) Low resistance in series
    (C) High resistance in parallel
    (D) Low resistance in parallel

  9. The path of a charged particle entering a uniform magnetic field at an angle other than 0°, 90° or 180° is a:
    (A) Straight line
    (B) Circle
    (C) Helix
    (D) Parabola

  10. A circular coil of radius R carrying current I has a magnetic dipole moment 'm'. If the radius is doubled and the current is halved, the new magnetic dipole moment will be:
    (A) m
    (B) 2m
    (C) m/2
    (D) 4m

Answers and Explanations:

  1. (C) Kinetic Energy: The magnetic force does no work on the charge, so its kinetic energy (and speed) remains constant. Velocity and momentum (vector quantities) change due to the change in direction.
  2. (B) Fleming's Left-Hand Rule: This rule is specifically used to find the direction of force on a current-carrying conductor in a magnetic field (or a moving charge).
  3. (C) Tesla (T): Tesla is the SI unit. Gauss is a CGS unit (1 T = 10⁴ G). Weber is the unit of magnetic flux.
  4. (A) 10⁻⁵ T: Using B = μ₀I / 2πa = (4π × 10⁻⁷ × 10) / (2π × 0.2) = (2 × 10⁻⁶) / 0.2 = 10 × 10⁻⁶ = 10⁻⁵ T.
  5. (D) Gauss's Law: Both relate a field (E or B) integrated over a closed surface/loop to the source (charge or current) enclosed. ∮ E ⋅ dS = Q_enc/ε₀ and ∮ B ⋅ dl = μ₀ I_enc.
  6. (A) Directly proportional to the current (I): Formula for solenoid field is B = μ₀ n I. So, B ∝ n and B ∝ I.
  7. (B) Attractive: Parallel currents attract, antiparallel currents repel.
  8. (D) Low resistance in parallel: A low resistance shunt is connected in parallel to divert most of the current, allowing the galvanometer to measure a much larger total current.
  9. (C) Helix: The component of velocity perpendicular to B causes circular motion, and the component parallel to B causes linear motion along the field lines. The combination results in a helical path.
  10. (B) 2m: Magnetic moment m = NIA = I (πR²). New moment m' = (I/2) * π(2R)² = (I/2) * π(4R²) = 2 * (I πR²) = 2m.

Remember to thoroughly revise these concepts and practice numerical problems based on the formulas. Good luck with your preparation!

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