🔄 Gauss’s Law

Gauss’s Theorem

∮ E⃗ ⋅ dA⃗ = Qₑₙcₗₒₛₑd/ε₀

Electric flux through closed surface

∝ enclosed charge

🎯 Applications of Gauss’s Law

⚪
Spherical Charge

E = kQ/r² (outside)

E = kQr/R³ (inside, uniform)

📏
Line Charge

E = λ/2πε₀r

Cylindrical Gaussian surface

📄
Sheet Charge

E = σ/2ε₀

Cylindrical Gaussian surface

⚡ Electric Potential

Definition of Electric Potential

V = W/q₀ = U/q₀

Work done per unit charge to bring from infinity

V = kQ/r (point charge)

Unit: Volt (V) = J/C

⚡ Potential Difference

Work and Potential Difference

V_AB = V_A – V_B = W_AB/q₀

W = q₀(V_A – V_B)

Work done in moving charge from B to A

🔗 Relationship Between E and V

Electric Field from Potential

E⃗ = -∇V = -dV/dr r̂

E = -dV/dr (radial field)

For uniform field: E = V/d

🗺️ Equipotential Surfaces

Properties of Equipotential Surfaces

⚡ Same potential everywhere on surface

🔄 No work needed to move charge along surface

⊥ Electric field always perpendicular to surface

⚡ Potential Due to Point Charges

Superposition of Potentials

V = V₁ + V₂ + V₃ + …

V = k∑(qᵢ/rᵢ)

Scalar addition (no vector addition needed)

🧲 Potential Due to Electric Dipole

Dipole Potential

V = (1/4πε₀) × (p⃗ ⋅ r̂)/r²

V = kp cos θ/r²

θ = angle between p⃗ and position vector

⚡ Potential Energy in Electric Field

Potential Energy Formulas

U = qV

U = kq₁q₂/r (two point charges)

U = (1/2)∑qᵢVᵢ (system of charges)

🔌 Conductors in Electrostatic Equilibrium

Properties of Conductors:

  • Electric field inside conductor = 0
  • All charges reside on the surface
  • Electric field just outside surface ⊥ to surface
  • Entire conductor is equipotential
  • Surface is equipotential
  • Charges redistribute to maintain equilibrium

⚡ Electric Field Near Conductor Surface

Field Just Outside Conductor

E = σ/ε₀

σ = local surface charge density

Field is perpendicular to surface

🛡️ Electrostatic Shielding

🧱 Dielectrics

🔄
Polar Molecules

Permanent dipole moment

H₂O, HCl

⚖️
Non-polar Molecules

Induced dipole moment

N₂, O₂, CO₂

Dielectric Constant

K = ε/ε₀ = E₀/E

K > 1 for all dielectrics

🔋 Capacitance

Definition of Capacitance

C = Q/V

Ability to store charge per unit potential

Unit: Farad (F) = C/V

📄 Parallel Plate Capacitor

Capacitance Formulas

C = ε₀A/d (vacuum)

C = Kε₀A/d (with dielectric)

A = area of plates, d = separation

K = dielectric constant

⚪ Capacitance of Different Geometries

GeometryCapacitance FormulaNotes
Parallel PlatesC = ε₀A/dMost common type
Concentric SpheresC = 4πε₀r₁r₂/(r₂-r₁)r₁ < r₂ (radii)
Isolated SphereC = 4πε₀RR = radius
Coaxial CylindersC = 2πε₀L/ln(b/a)L = length, a < b

🔗 Combination of Capacitors

📏
Series

1/C = 1/C₁ + 1/C₂ + …

Same charge, different voltages

🔀
Parallel

C = C₁ + C₂ + …

Same voltage, different charges

⚡ Energy Stored in Capacitor

Energy Storage Formulas

U = ½CV² = ½QV = ½Q²/C

Energy density = ½ε₀E²

Energy stored in electric field

⚡ Van de Graaff Generator

Working Principle

🔄 Moving belt transfers charge to metal dome

⚡ High voltage (up to millions of volts)

🛡️ Dome acts as Faraday cage

📝 Key Formulas Summary

Coulomb’s Law: F = kq₁q₂/r²
Electric Field: E = F/q₀ = kQ/r²
Electric Potential: V = kQ/r
Gauss’s Law: ∮E⃗⋅dA⃗ = Q/ε₀
E-V Relation: E = -dV/dr
Capacitance: C = Q/V = ε₀A/d
Energy: U = ½CV² = ½QV
Dipole Moment: p = q × 2a

📊 Important Constants

ConstantSymbolValueUnit
Coulomb’s constantk9 × 10⁹Nm²/C²
Permittivity of free spaceε₀8.85 × 10⁻¹²C²/Nm²
Elementary chargee1.6 × 10⁻¹⁹C
Electron massmₑ9.1 × 10⁻³¹kg
Proton massmₚ1.67 × 10⁻²⁷kg

💡 Problem Solving Strategy

⚠️ Common Mistakes to Avoid

🔬 Real-World Applications

🎯 Exam Preparation Tips

📚
Theory

Understand field line concepts

Learn conductor properties

🧮
Calculations

Practice Coulomb’s law problems

Capacitor combinations

📊
Concepts

Gauss’s law applications

E and V relationships

🔬
Applications

Van de Graaff generator

Electrostatic shielding

🎊 Summary

⚡ Master the Electric World! ⚡

Understanding electrostatics is key to all electromagnetic phenomena!

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Electrostatics and Electric Potential: Class 12 Physics NCERT
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⚡

Electrostatics and Electric Potential

Class 12 Physics – NCERT Chapter 1 & 2

Explore the World of Static Electric Charges

🌟 Introduction to Electrostatics

  • Study of electric charges at rest
  • Fundamental force of nature – electromagnetic force
  • Ancient Greeks discovered static electricity (amber – elektron)
  • Benjamin Franklin’s contributions to understanding electricity
  • Foundation for understanding electromagnetic phenomena
  • Applications: Van de Graaff generator, photocopying, lightning rods

⚡ Electric Charge

Fundamental Properties of Charge

Charge is quantized: Q = ±ne

e = 1.6 × 10⁻¹⁹ C (elementary charge)

➕
Positive Charge

Protons, glass rubbed with silk

Deficiency of electrons

➖
Negative Charge

Electrons, plastic rubbed with fur

Excess of electrons

📐 Properties of Electric Charge

  • Quantization: Charge exists in discrete packets (Q = ne)
  • Conservation: Total charge in isolated system remains constant
  • Additivity: Total charge = algebraic sum of individual charges
  • Invariance: Charge is independent of motion (relativistic invariant)
  • Like charges repel, unlike charges attract
  • Charge is transferable by conduction, induction, friction

🔌 Conductors and Insulators

🔗
Conductors

Free electrons available

Metals, graphite, ionized gases

🚫
Insulators

No free electrons

Glass, rubber, plastic, wood

⚡
Semiconductors

Intermediate conductivity

Silicon, germanium

🔄 Methods of Charging

🤝
Friction

Rubbing two materials

Electron transfer occurs

👋
Conduction

Direct contact with charged body

Charge redistribution

🪄
Induction

Without direct contact

Charge separation occurs

⚖️ Coulomb’s Law

Force Between Point Charges

F = k(q₁q₂)/r²

F = (1/4πε₀)(q₁q₂)/r²

k = 9 × 10⁹ Nm²/C² (Coulomb’s constant)

ε₀ = 8.85 × 10⁻¹² C²/Nm² (permittivity of free space)

  • Force ∝ product of charges (q₁q₂)
  • Force ∝ 1/r² (inverse square law)
  • Force acts along line joining the charges
  • Attractive if charges have opposite signs
  • Repulsive if charges have same signs

➕ Principle of Superposition

Total Force on a Charge

F⃗ = F⃗₁ + F⃗₂ + F⃗₃ + …

Vector sum of individual forces

  • Force on a charge due to multiple charges = vector sum
  • Forces due to different charges are independent
  • Each force calculated using Coulomb’s law separately
  • Final force found by vector addition
  • Applicable to any number of point charges

🌐 Electric Field

Electric Field Definition

E⃗ = F⃗/q₀

Force per unit positive test charge

E = kQ/r² (point charge)

Unit: N/C or V/m

  • Vector quantity with magnitude and direction
  • Independent of test charge q₀
  • Direction: same as force on positive test charge
  • Exists whether test charge is present or not

📈 Electric Field Lines

Properties of Field Lines

➡️ Direction: Tangent gives field direction

🔢 Density: Closer lines = stronger field

🚫 Never intersect or form closed loops

  • Start from positive charges, end on negative charges
  • Never cross each other
  • Density indicates field strength
  • Continuous curves in space
  • Tangent at any point gives field direction

⚡ Electric Field of Various Charge Configurations

ConfigurationElectric Field FormulaNotes
Point ChargeE = kQ/r²Radial field
Infinite Line ChargeE = λ/2πε₀rλ = linear charge density
Infinite SheetE = σ/2ε₀σ = surface charge density
Uniformly Charged RingE = kQx/(x²+R²)^(3/2)On axis, x from center
Electric DipoleE = kp/r³ (axial)p = dipole moment

🧲 Electric Dipole

Electric Dipole Moment

p⃗ = q × 2a⃗

p = magnitude of dipole moment

2a = separation between charges

Direction: from -q to +q

  • System of two equal and opposite charges
  • Dipole moment: vector from negative to positive charge
  • Unit: C⋅m (coulomb meter)
  • Important in molecular physics and chemistry

⚡ Dipole in Uniform Electric Field

Torque and Energy

τ⃗ = p⃗ × E⃗ = pE sin θ

U = -p⃗ ⋅ E⃗ = -pE cos θ

θ = angle between p⃗ and E⃗

  • Net force on dipole in uniform field = 0
  • Torque tends to align dipole with field
  • Minimum energy when θ = 0° (aligned)
  • Maximum energy when θ = 180° (anti-aligned)
  • Equilibrium positions: θ = 0° (stable), θ = 180° (unstable)

🔄 Gauss’s Law

Gauss’s Theorem

∮ E⃗ ⋅ dA⃗ = Qₑₙcₗₒₛₑd/ε₀

Electric flux through closed surface

∝ enclosed charge

  • Relates electric flux to enclosed charge
  • Fundamental law of electrostatics

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