Class 12 Physics Notes Chapter 1 (Electric Charges and Fields) – Examplar Problems (English) Book

Alright class, let's get straight into the crucial concepts of Chapter 1, 'Electric Charges and Fields', focusing on what you need for your competitive government exams, keeping the NCERT Exemplar perspective in mind. This chapter forms the bedrock of electrostatics.

Chapter 1: Electric Charges and Fields - Detailed Notes for Exam Preparation

1. Electric Charge:

• Definition: Intrinsic property of elementary particles of matter which gives rise to electric force between various objects.
• Types: Positive (+) and Negative (-). Like charges repel, unlike charges attract.
• Unit: SI unit is Coulomb (C). Dimensional Formula: [AT].
• Properties:
  • Additivity: Total charge on a body is the algebraic sum of all individual charges present. (Scalar addition).
  • Conservation: The total charge of an isolated system remains constant. Charge can neither be created nor destroyed, only transferred. (Crucial for reaction-based questions).
  • Quantization: Charge exists in discrete packets rather than continuous amounts. The smallest unit of free charge is the charge of an electron/proton (e = 1.602 × 10⁻¹⁹ C). Total charge (Q) on a body is always an integral multiple of 'e'. Q = ± ne, where 'n' is an integer (1, 2, 3,...). Exemplar often tests conceptual understanding here - can a body have a charge of 2.5e? No.
  • Invariance: Charge on a body does not depend on its speed (unlike mass).

2. Methods of Charging:

• Friction: Rubbing two suitable materials transfers electrons from one to the other.
• Conduction: Charging an uncharged conductor by bringing it into contact with a charged conductor. Charge is shared.
• Induction: Charging an object without direct contact. A charged object brought near an uncharged conductor causes charge separation. Earthing can then be used to remove one type of charge, leaving the conductor charged. Important for conceptual MCQs.

3. Coulomb's Law:

• Statement: The electrostatic force between two point charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. It acts along the line joining the two charges.
• Mathematical Form:
  F = k * |q₁q₂| / r²
  where:
  • F is the magnitude of the force.
  • q₁, q₂ are the magnitudes of the point charges.
  • r is the distance between the charges.
  • k is Coulomb's constant = 1 / (4πε₀) ≈ 9 × 10⁹ Nm²/C².
  • ε₀ is the permittivity of free space = 8.854 × 10⁻¹² C²/Nm².
• Vector Form: Essential for direction and superposition.
  Force on q₂ due to q₁: F₂₁ = k * q₁q₂ / |r₂₁|³ * r₂₁ where r₂₁ = r₂ - r₁ is the position vector from q₁ to q₂.
  Note: F₂₁ = - F₁₂ (Obeys Newton's Third Law).
• Effect of Medium: If charges are placed in a medium with relative permittivity εr (or dielectric constant K), the force becomes:
  F_medium = F_vacuum / K = F_vacuum / εr = (1 / (4πε₀εr)) * |q₁q₂| / r²
  where ε = ε₀εr is the permittivity of the medium. Force decreases in a medium.
• Principle of Superposition: The net force on any charge due to a number of other charges is the vector sum of all the forces exerted on that charge by the other charges, taken one at a time.
  F_net = F₁ + F₂ + F₃ + ... (Vector addition!)

4. Electric Field (E):

• Concept: The region around a charged body within which its influence (electrostatic force) can be experienced by another charge.
• Electric Field Intensity (E): Force experienced by a unit positive test charge (q₀) placed at that point, without disturbing the source charge.
  E = F / q₀
• Unit: N/C or V/m. Dimensional Formula: [MLT⁻³A⁻¹]. Vector quantity.
• Direction: Direction of the force that would act on a positive test charge. Radially outward from a positive source charge, radially inward towards a negative source charge.
• Electric Field due to a Point Charge (q):
  E = k * q / r² * r̂ (where r̂ is the unit vector from q to the point). Magnitude: E = k * |q| / r².
• Superposition Principle for Electric Fields: Net electric field at a point due to multiple charges is the vector sum of the electric fields due to individual charges.
  E_net = E₁ + E₂ + E₃ + ... (Vector addition!)

5. Electric Field Lines:

• Definition: Imaginary lines or curves drawn in an electric field such that the tangent at any point gives the direction of the electric field intensity at that point.
• Properties: (Very important for conceptual questions)
  • Start from positive charges, end at negative charges (or infinity).
  • Tangent at any point gives the direction of E.
  • They never intersect each other (because E has a unique direction at any point).
  • Relative density of field lines indicates the strength of the electric field (closer lines = stronger field).
  • They are always normal (perpendicular) to the surface of a conductor in electrostatic equilibrium.
  • They do not form closed loops (electrostatic field is conservative).
  • They tend to contract longitudinally (explains attraction) and exert lateral pressure (explains repulsion).

6. Electric Dipole:

• Definition: A system of two equal and opposite charges (+q and -q) separated by a small distance (usually 2a).
• Dipole Moment (p): Measure of the strength of the electric dipole.
  p = q * (2a)
  • Vector quantity. Direction is from the negative charge (-q) to the positive charge (+q).
  • Unit: Coulomb-meter (Cm).
• Electric Field due to a Dipole:
  • On the Axial Line (End-on position): At a distance r from the center (r >> a):
    E_axial = (1 / 4πε₀) * (2p / r³) (Direction along p)
  • On the Equatorial Line (Broadside-on position): At a distance r from the center (r >> a):
    E_equatorial = (1 / 4πε₀) * (p / r³) (Direction opposite to p)
  • Note: E_axial = 2 * E_equatorial for the same distance r.
• Torque on a Dipole in a Uniform Electric Field (E):
  τ = p × E Magnitude: τ = pE sinθ
  where θ is the angle between p and E. Torque is maximum when θ = 90°, zero when θ = 0° (stable equilibrium) or θ = 180° (unstable equilibrium).
• Potential Energy of a Dipole in a Uniform Electric Field (E):
  U = -p ⋅ E = -pE cosθ
  Minimum energy (most stable) at θ = 0° (U = -pE). Maximum energy (most unstable) at θ = 180° (U = +pE). Zero energy at θ = 90°.

7. Electric Flux (Φ):

• Definition: A measure of the total number of electric field lines passing normally through a given area.
• Mathematical Form: For a small area element dA in an electric field E: dΦ = E ⋅ dA = E dA cosθ
  Total flux through a surface S: Φ = ∫_S E ⋅ dA
• Unit: Nm²/C or Vm. Scalar quantity.
• θ is the angle between the electric field vector E and the area vector dA (which is normal to the surface).

8. Gauss's Law:

• Statement: The total electric flux through any closed surface (called a Gaussian surface) is equal to 1/ε₀ times the net charge enclosed by the surface.
  Φ_closed = ∮ E ⋅ dA = q_enc / ε₀
• Key Points:
  • Applies to any closed surface.
  • q_enc is the net charge (algebraic sum) inside the surface. Charges outside do not contribute to the net flux, although they might affect the electric field E at the surface.
  • Extremely useful for calculating electric fields for symmetric charge distributions.
  • Gaussian surface is an imaginary surface chosen strategically to simplify calculations (often exploiting symmetry).
• Applications (Important Derivations & Results):
  • Field due to an infinitely long straight uniformly charged wire: E = λ / (2πε₀r) (where λ is linear charge density, r is perpendicular distance). E ∝ 1/r.
  • Field due to a uniformly charged infinite plane sheet: E = σ / (2ε₀) (where σ is surface charge density). E is independent of distance from the sheet.
  • Field due to a uniformly charged thin spherical shell (Charge Q, Radius R):
    • Outside (r > R): E = kQ / r² (Acts like a point charge at the center).
    • On the surface (r = R): E = kQ / R².
    • Inside (r < R): E = 0. (Crucial result!)
  • Field due to a uniformly charged non-conducting solid sphere (Charge Q, Radius R):
    • Outside (r > R): E = kQ / r².
    • On the surface (r = R): E = kQ / R².
    • Inside (r < R): E = (kQr) / R³ = (ρr) / (3ε₀) (where ρ is volume charge density). E ∝ r.

9. Conductors in Electrostatic Equilibrium:

• Electric field inside a conductor is zero.
• Electric field just outside a charged conductor is perpendicular to the surface and has magnitude E = σ / ε₀ (where σ is the local surface charge density).
• Net charge inside a conductor is zero; any excess charge resides on its surface.
• Electric potential is constant throughout the volume of the conductor and equal to the value on its surface.

Practice MCQs (Based on Exemplar Style):

1. A metallic sphere is placed in a uniform electric field. Which path correctly represents the electric field lines?
   (a) Straight lines passing through the sphere.
   (b) Lines bending towards the center of the sphere.
   (c) Lines terminating and originating perpendicularly on the surface of the sphere.
   (d) Lines forming closed loops around the sphere.

2. Two point charges +q and -q are placed at distance d. If a third charge +Q is placed exactly midway between them, the net force on +Q is:
   (a) Directed towards +q
   (b) Directed towards -q
   (c) Zero
   (d) Cannot be determined

3. The electric flux through a closed surface enclosing a net charge q is Φ. If the surface is replaced by a larger closed surface that still encloses the charge q, the electric flux will be:
   (a) Φ/2
   (b) Φ
   (c) 2Φ
   (d) Zero

4. An electric dipole of moment p is placed in a uniform electric field E. The maximum torque experienced by the dipole is:
   (a) pE
   (b) Zero
   (c) 2pE
   (d) pE/2

5. A thin spherical shell of radius R has charge Q uniformly distributed on its surface. The electric field at a point inside the shell (r < R) is:
   (a) kQ/R²
   (b) kQ/r²
   (c) Zero
   (d) kQr/R³

6. Which of the following properties is NOT true for electric field lines?
   (a) They start from positive charges and end at negative charges.
   (b) Two field lines can never cross each other.
   (c) They form closed loops.
   (d) The tangent to a field line gives the direction of the electric field.

7. When a glass rod is rubbed with silk, the glass rod acquires a positive charge because:
   (a) Protons are added to it.
   (b) Electrons are added to it.
   (c) Protons are removed from it.
   (d) Electrons are removed from it.

8. The force between two point charges in vacuum is F. If a dielectric medium of dielectric constant K=4 is introduced between them, the new force will be:
   (a) 4F
   (b) F
   (c) F/4
   (d) F/2

9. Consider a system of two charges +q and +q separated by a distance r. The electric field is zero at a point:
   (a) Midway between the charges.
   (b) At a distance r/4 from either charge.
   (c) At infinity.
   (d) No finite point exists where the field is zero.

10. A charge Q is placed at the corner of a cube. The electric flux through all the six faces of the cube is:
    (a) Q/ε₀
    (b) Q/(2ε₀)
    (c) Q/(6ε₀)
    (d) Q/(8ε₀)

Answers to MCQs:

1. (c) - Field lines are perpendicular to conductor surfaces and E=0 inside.
2. (b) - Force due to +q is repulsive, force due to -q is attractive. Both forces are equal in magnitude and point towards -q.
3. (b) - Gauss's law states flux depends only on the enclosed charge, not the size/shape of the Gaussian surface.
4. (a) - Torque τ = pE sinθ. Maximum when sinθ = 1 (θ=90°).
5. (c) - Standard result from Gauss's law application.
6. (c) - Electrostatic field lines do not form closed loops.
7. (d) - Charging by friction involves transfer of electrons. Glass loses electrons to silk.
8. (c) - F_medium = F_vacuum / K.
9. (a) - For two equal positive charges, the fields cancel exactly midway between them.
10. (d) - The charge at the corner is shared by 8 identical cubes. The flux through one cube is (1/8)th of the total flux (Q/ε₀) that would emanate if the charge were fully enclosed.

Make sure you thoroughly understand these concepts, especially the vector nature of forces and fields, superposition, Gauss's law applications, and dipole behavior. Practice numerical problems and conceptual questions from the Exemplar book itself. Good luck!