Moving Charges and Magnetism Class 12 – NCERT Concepts, Formulas & CBSE Board Questions

 

Here are some important pointers drawn from the provided sources regarding moving charges and magnetism:

• Historical Context and Fundamental Connection:

  • Electricity and magnetism have been known for over 2000 years, but their intimate relationship was discovered only about 200 years ago, in 1820, by Danish physicist Hans Christian Oersted.
  • Oersted observed that an electric current in a straight wire caused a noticeable deflection in a nearby magnetic compass needle, demonstrating that moving charges or currents produce a magnetic field in the surrounding space.
  • This discovery led to intense experimentation, culminating in James Maxwell unifying the laws of electricity and magnetism in 1864, realizing that light itself was made of electromagnetic waves.

• Magnetic Field (B):

  • Just as static charges produce an electric field, currents or moving charges produce a magnetic field, denoted by B.
  • The magnetic field is a vector field defined at each point in space and obeys the principle of superposition, meaning the total magnetic field from multiple sources is the vector sum of individual fields.
  • The SI unit of magnetic field is tesla (T), named after Nikola Tesla. A smaller non-SI unit called gauss is also often used, where 1 gauss = 10⁻⁴ tesla.
  • The earth's magnetic field is approximately 3.6 × 10⁻⁵ T.
  • In diagrams, a dot (¤) indicates a current or field emerging out of the plane of the paper, while a cross () indicates one going into the plane.

• Lorentz Force:

  • The total force (Lorentz force) on a point charge q moving with velocity v in the presence of both an electric field E and a magnetic field B is given by F = q [E(r) + v × B(r)]. This force was first given by H.A. Lorentz.
  • Key features of the magnetic force (q [v × B]):
    • It depends on the charge (q), velocity (v), and magnetic field (B).
    • The force on a negative charge is opposite to that on a positive charge.
    • It includes a vector product (cross product) of velocity and magnetic field, meaning the force acts in a direction perpendicular to both the velocity and the magnetic field. Its direction is given by the screw rule or right-hand rule.
    • The magnetic force vanishes if the velocity and magnetic field are parallel or anti-parallel.
    • The magnetic force is zero if the charge is not moving (|v|=0); only a moving charge feels the magnetic force.
    • No work is done by the magnetic force on a moving charge, and thus no change in the magnitude of the velocity is produced, only the direction of momentum may change.

• Magnetic Force on a Current-Carrying Conductor:

  • A straight conductor of length l carrying a steady current I experiences a force F in a uniform external magnetic field B, given by F = I l × B. The vector l has a magnitude equal to the length and a direction identical to the current.

• Motion of a Charged Particle in a Magnetic Field:

  • If a charged particle's velocity v is perpendicular to a uniform magnetic field B, the magnetic force acts as a centripetal force, causing the particle to move in a circular motion perpendicular to the magnetic field.
  • The radius (r) of this circular path is given by r = mv / qB.
  • The angular frequency (ω) or frequency of rotation (ν), known as the cyclotron frequency, is independent of the particle's speed or energy (ω = qB/m, ν = qB/(2πm)). This independence is crucial for the design of cyclotrons, which accelerate charged particles.
  • If the velocity has a component parallel to the magnetic field, the motion along the field remains unaffected, resulting in a helical motion.

• Biot-Savart Law:

  • This law describes the magnetic field dB produced by an infinitesimal current element I dl at a point P at a distance r from the element.
  • In vector notation, dB = (μ₀/4π) * (I dl × r) / r³.
  • The magnitude is given by dB = (μ₀/4π) * (I dl sinθ) / r², where θ is the angle between dl and r.
  • The direction of dB is perpendicular to the plane containing dl and r, given by the right-hand screw rule.
  • Permeability of free space (μ₀) is a constant of proportionality, with an exact value of 4π × 10⁻⁷ T m A⁻¹ in SI units.
  • Similarities with Coulomb's Law: Both are long-range laws (inverse square dependence on distance), and the principle of superposition applies to both fields.
  • Differences from Coulomb's Law: Electrostatic fields are produced by scalar sources (electric charge), while magnetic fields are produced by vector sources (I dl). The electrostatic field is along the displacement vector, but the magnetic field is perpendicular to the plane containing the displacement vector and current element. The Biot-Savart law also has an angle dependence.

• Magnetic Field Due to Specific Configurations:

  • Circular Current Loop:
    • On the axis at distance x from the center: B = (μ₀IR² / 2(x²+R²)^3/2).
    • At the center of the loop (x=0): B = μ₀I / 2R.
    • The right-hand thumb rule for circular wires states: Curl the palm of your right hand around the circular wire with fingers pointing in the direction of the current; your right-hand thumb gives the direction of the magnetic field.
    • The magnetic field lines form closed loops.
  • Long Straight Wire:
    • The magnetic field at a distance r from a long, straight wire carrying current I is B = μ₀I / (2πr).
    • The field direction at any point on a circle around the wire is tangential to it, forming concentric circles.
    • The right-hand rule for a straight wire states: Grasp the wire in your right hand with your extended thumb pointing in the direction of the current; your fingers will curl around in the direction of the magnetic field.
  • Solenoid:
    • A solenoid is a long wire wound in a helix.
    • For a long solenoid, the magnetic field inside is uniform, strong, and parallel to the axis, given by B = μ₀nI, where n is the number of turns per unit length.
    • The field outside a long solenoid approaches zero.

• Ampere's Circuital Law:

  • This law provides an alternative way to express the Biot-Savart law and is analogous to Gauss's law in electrostatics.
  • It states that the integral of the tangential component of the magnetic field B around a closed loop (Amperian loop) is equal to μ₀ times the total current I passing through the surface bounded by the loop: ∫ B.dl = μ₀I.
  • A sign convention is used: if the fingers of the right hand are curled in the sense the boundary is traversed, the thumb gives the sense in which the current is positive.
  • For cases with sufficient symmetry (e.g., a long straight wire), the law simplifies to BL = μ₀I_e, where L is the length of the loop where B is tangential and I_e is the enclosed current.
  • Ampere's law holds for steady currents that do not fluctuate with time.

• Force Between Two Parallel Currents:

  • Two parallel current-carrying conductors exert magnetic forces on each other.
  • Parallel currents attract each other, while anti-parallel currents repel each other. This is opposite to electrostatics where like charges repel.
  • The magnitude of the force per unit length (f) between two long parallel wires separated by distance d and carrying currents I_a and I_b is f = μ₀I_a I_b / (2πd).
  • This expression is used to define the ampere (A), one of the seven SI base units. One ampere is the steady current that, when maintained in two very long, straight, parallel conductors of negligible cross-section, placed one meter apart in vacuum, would produce a force of 2 × 10⁻⁷ newtons per meter of length on each conductor.
  • The coulomb (C), the SI unit of charge, is then defined in terms of the ampere: 1 coulomb is the quantity of charge that flows through a conductor's cross-section in 1 second when a steady current of 1A is set up.

• Torque on a Current Loop and Magnetic Dipole:

  • A rectangular loop carrying a steady current I in a uniform magnetic field experiences a torque but no net force. This is analogous to an electric dipole in a uniform electric field.
  • The magnitude of the torque (τ) on a loop with area A carrying current I in a uniform magnetic field B is given by τ = IABsinθ.
  • The magnetic moment (m) of a current loop is defined as m = IA. For a coil with N closely wound turns, m = NIA. The direction of the area vector A (and thus m) is given by the right-hand thumb rule (curling fingers in current direction, thumb points to m).
  • Using magnetic moment, the torque can be expressed as a vector product: τ = m × B.
  • The torque vanishes when the magnetic moment m is parallel or anti-parallel to the magnetic field B, indicating equilibrium. Stable equilibrium occurs when m and B are parallel.
  • A circular current loop behaves like a magnetic dipole at large distances, analogous to an electric dipole. A key difference is that magnetic monopoles are not known to exist, unlike electric charges which form electric dipoles.

• Moving Coil Galvanometer (MCG):

  • The MCG is a device used to measure currents and voltages.
  • Its principle is based on the torque experienced by a current-carrying coil in a magnetic field.
  • A spring provides a counter-torque, balancing the magnetic torque (NIAB) and resulting in a steady angular deflection (φ). In equilibrium, kφ = NIAB, where k is the torsional constant of the spring.
  • Current sensitivity (deflection per unit current) is φ/I = NAB/k. Increasing the number of turns N increases current sensitivity.
  • Conversion to Ammeter: To measure current, a galvanometer, which is highly sensitive and has large resistance, is connected in series. To prevent it from changing the circuit current significantly and to allow it to measure larger currents, a small resistance called a shunt resistance (r_s) is connected in parallel with the galvanometer coil. Most of the current then passes through the shunt.
  • Conversion to Voltmeter: To measure voltage across a section, a galvanometer must be connected in parallel and draw very little current. To achieve this, a large resistance (R) is connected in series with the galvanometer.
  • Voltage sensitivity (deflection per unit voltage) is φ/V = NAB/(kR). Note that increasing current sensitivity (e.g., by doubling N) does not necessarily increase voltage sensitivity, as the galvanometer's resistance also increases.

📘 Chapter 4: Moving Charges and Magnetism (Class 12 CBSE)

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🔑 Key Concepts & Theorems

1. Lorentz Force

   • Force on a moving charge in magnetic and electric fields:

     $$\vec{F} = q(\vec{v} \times \vec{B})$$

2. Motion in Magnetic Field

   • Circular path if velocity ⊥ field:

     $$r = \frac{mv}{qB}$$

   • Pitch of helix:

     $$p = v_{\parallel}T$$

3. Force on Current-Carrying Conductor

   $$\vec{F} = I(\vec{L} \times \vec{B})$$

4. Biot–Savart Law

   • Magnetic field due to a current element:

     $$d\vec{B} = \frac{\mu_0}{4\pi} \cdot \frac{I d\vec{l} \times \vec{r}}{r^3}$$

5. Magnetic Field due to:

   • Straight Wire:

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

   • Circular Loop (at center):

     $$B = \frac{\mu_0 I}{2R}$$

   • Solenoid (long):

     $$B = \mu_0 n I$$

6. Ampere’s Circuital Law

   $$\oint \vec{B} \cdot d\vec{l} = \mu_0 I_\text{enc}$$

7. Force between Two Parallel Currents

   $$F/l = \frac{\mu_0 I_1 I_2}{2\pi d}$$

   • Basis of definition of ampere

8. Torque on a Current Loop

   $$\tau = nBIA \sin\theta$$

9. Magnetic Dipole Moment

   $$M = I \cdot A$$

10. Moving Coil Galvanometer

    • Principle: Current loop in magnetic field experiences torque

    • Deflection:

      $$\theta \propto I$$

    • Used to measure small currents

11. Conversion of Galvanometer

    • To Ammeter: Connect low shunt resistance in parallel

    • To Voltmeter: Connect high resistance in series

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📗 NCERT Exercise Questions with Solutions

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Q1. A charged particle enters a magnetic field at right angle. Describe its motion.

Answer:
The particle will move in a circular path due to centripetal force provided by the magnetic Lorentz force.

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Q2. Derive the expression for magnetic force on a current-carrying conductor.

Answer:
From Lorentz force:

$$\vec{F} = q(\vec{v} \times \vec{B})$$

For current:

$$\vec{F} = I(\vec{L} \times \vec{B})$$

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Q3. Use Biot–Savart law to derive magnetic field at center of circular loop.

Answer:

$$B = \frac{\mu_0 I}{2R}$$

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Q4. Derive magnetic field inside a long straight solenoid using Ampere’s Law.

Answer:
Using:

$$\oint \vec{B} \cdot d\vec{l} = \mu_0 n I l \Rightarrow B = \mu_0 n I$$

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Q5. What is the force between two parallel current-carrying wires?

Answer:

$$F/l = \frac{\mu_0 I_1 I_2}{2\pi d}$$

Attractive if currents are in same direction.

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Q6. Explain construction and working of a moving coil galvanometer.

Answer:

• Coil suspended in magnetic field

• Torque causes deflection:

  $$\tau = nBIA \Rightarrow \theta \propto I$$

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📋 CBSE PYQs – Moving Charges and Magnetism

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🔹 Q (2023)

Derive expression for magnetic field inside long solenoid. What are its properties?

Answer:

$$B = \mu_0 n I$$

• Uniform field

• Directed along axis

• Independent of solenoid diameter

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🔹 Q (2022)

Explain force between two parallel current-carrying wires. Define 1 Ampere.

Answer:

$$F/l = \frac{\mu_0 I_1 I_2}{2\pi d}$$

If $F/l = 2 \times 10^{-7} \, N/m$ when $I_1 = I_2 = 1A$, $d = 1m$ → defines 1 ampere.

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🔹 Q (2021)

Derive torque on a rectangular coil in uniform magnetic field.

Answer:

$$\tau = nBIA \sin\theta$$

Used in galvanometers.

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🔹 Q (2020)

Describe conversion of galvanometer into ammeter and voltmeter.

Answer:

• Ammeter: Shunt in parallel

  $$R_\text{sh} = \frac{Ig Rg}{I - Ig}$$

• Voltmeter: Resistance in series

  $$R_s = \frac{V}{Ig} - Rg$$

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