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Electric Charges and Fields Overview

Electric Field Magnitude

• Definition: The electric field E(r) at a point depends on the charge distribution.

• Spherical Gaussian Surface:

  • Charged nucleus and surrounding sphere of radius r contribute to total charge q enclosed.

  • Electric field direction: Radially outward when r < R.

• Application of Gauss's Law:

  • For r < R:

    • Enclosed charge = Positive nuclear charge + Negative charge within sphere;

    • ( E(r) = \frac{Z e}{4 \pi \epsilon_0 r^2} ) ( (r < R) )

Electric Charge Properties

1. Two Types of Charges: Like charges repel; unlike charges attract.

   • Example: Glass rod (positive) rubbed with silk; plastic rod (negative) rubbed with fur.

2. Conductors vs Insulators:

   • Conductors: Allow charge movement (e.g., metals: mobile electrons).

   • Insulators: Prevent charge movement.

3. Basic Properties of Electric Charge:

   • Quantisation: q = n * e (n = integer), where e = elementary charge.

   • Additivity: Total charge = Algebraic sum of individual charges.

   • Conservation: Total charge in an isolated system remains constant.

Coulomb’s Law

• Formula: ( F_{21} = k \frac{q_1 q_2}{r^2} )

  • ( k = \frac{1}{4 \pi \epsilon_0} )

• SI Unit of Charge: Coulomb (C).

• Key Relationships:

  • Ratio of electric to gravitational force: ( \frac{F_{e}}{F_{g}} \approx \frac{2.39 \times 10^{39} e^2}{G m_p m_e} )

Superposition Principle

• Forces between charges add vectorially; individual forces are calculated via Coulomb’s Law.

Electric Field Definition

• Concept: Electric field E at a point due to charge configuration is defined as the force on a small positive test charge divided by its magnitude.

• For a point charge:

  • ( |E| = \frac{|q|}{4 \pi \epsilon_0 r^2} )

  • Radial direction depends on charge sign.

Electric Field Lines

• Characteristics:

  1. Continuous curves with no breaks.

  2. Cannot intersect each other.

  3. Start at positive charges and end at negative charges.

• Strength Representation: Closer lines indicate stronger fields.

Electric Dipole

• Definition: Pair of charges (+q, -q) separated by distance 2a.

• Dipole Moment: ( p = 2qa ) directed from -q to +q.

Electric Field of a Dipole

• In Equatorial Plane: ( E = \frac{1}{4 \pi \epsilon_0} \frac{p}{r^3} )

• On Axis: ( E = \frac{1}{4 \pi \epsilon_0} \frac{2p}{r^3} )

• Notable: Dipole field decreases with ( 1/r^3 ) unlike point charge's ( 1/r^2 ).

Electric Field and Torque on a Dipole

• In uniform electric fields:

  • Torque: ( \tau = p \times E )

  • No net force is experienced by the dipole.

Gauss’s Law

• Mathematical Statement: ( \Phi_E = \frac{Q_{enc}}{ ho_0} )

• Applications:

  1. Infinitely Long Straight Wire: ( E = \frac{\lambda}{2 \pi \epsilon_0 r} )

  2. Infinite Plane Sheet: ( E = \frac{\sigma}{2 \epsilon_0} )

  3. Thin Spherical Shell:

     • Outside: ( E = \frac{q}{4 \pi \epsilon_0 r^2} )

     • Inside: ( E = 0 )

Properties of Electric Quantities

1. Vector Area Element: ( DS [L^2] = m^2 )

2. Electric Field: ( E [MLT^{-3}A^{-1}] = V m^{-1} )

3. Electric Flux: ( \Phi_E [ML^3 T^{-3}A^{-1}] = V m )

4. Dipole Moment: ( p [LTA] = C m )

5. Charge Densities: Linear (C/m), surface (C/m²), volume (C/m³) charges.

Points to Ponder

1. Protons in Nucleus: Held by a strong nuclear force effective over short distances (approx. 10^-14 m).

2. Differences in Forces: Coulomb's law applies to electrostatic force (attractive or repulsive), while gravitational force is solely attractive.