Electric Charges and Fields

• Electric Charge

  • Concept: Intrinsic property of matter causing electrostatic force
  • Types: Positive (+ve) and Negative (-ve)
  • Basic Properties
    • Additivity: Total charge $Q = \sum q_i$
    • Quantization: $q = \pm ne$ (where $e = 1.6 \times 10^{-19}\text{ C}$)
    • Conservation: Net charge of isolated system remains constant
    • Invariance: Charge value is independent of frame/speed
  • Methods of Charging: Friction, Conduction, Induction

• Coulomb's Law

  • Statement: Force between two stationary point charges
  • Formula: $F = \frac{1}{4\pi\varepsilon_0} \frac{|q_1 q_2|}{r^2}$
  • Electrostatic Constant ($k$): $k = \frac{1}{4\pi\varepsilon_0} \approx 9 \times 10^9\text{ N}\cdot\text{m}^2/\text{C}^2$
  • Permittivity
    • Free Space ($\varepsilon_0$): $8.854 \times 10^{-12}\text{ C}^2/\text{N}\cdot\text{m}^2$
    • Dielectric Constant ($K$ or $\varepsilon_r$): $\varepsilon_r = \frac{\varepsilon}{\varepsilon_0} = \frac{F_{\text{vacuum}}}{F_{\text{medium}}}$
  • Superposition Principle: Net force on a charge is the vector sum of all individual forces

• Electric Field (EF)

  • Concept: Region around a charge where electric force is experienced
  • Electric Field Intensity ($\vec{E}$)
    • Definition: Force per unit positive test charge ($\vec{E} = \frac{\vec{F}}{q_0}$)
    • Unit & Dimension: $\text{N/C}$ or $\text{V/m}$, $[M^1 L^1 T^{-3} A^{-1}]$
    • Point Charge Field: $E = \frac{1}{4\pi\varepsilon_0} \frac{q}{r^2}$
  • Electric Field Lines
    • Originates from +ve charge, terminates at -ve charge
    • Do not form closed loops
    • Never intersect each other
    • Tangent gives direction of $\vec{E}$

• Electric Dipole

  • Definition: Pair of equal and opposite charges separated by small distance ($2a$)
  • Dipole Moment ($\vec{p}$)
    • Formula: $\vec{p} = q \times 2\vec{a}$
    • Direction: From -ve to +ve charge
    • Unit: $\text{C}\cdot\text{m}$
  • Electric Field of Dipole
    • Axial Position: $E_{\text{axial}} = \frac{1}{4\pi\varepsilon_0} \frac{2pr}{(r^2 - a^2)^2} \approx \frac{1}{4\pi\varepsilon_0} \frac{2p}{r^3}$
    • Equatorial Position: $E_{\text{eq}} = \frac{1}{4\pi\varepsilon_0} \frac{p}{(r^2 + a^2)^{3/2}} \approx \frac{1}{4\pi\varepsilon_0} \frac{p}{r^3}$
  • Dipole in Uniform Electric Field
    • Net Force: Zero
    • Torque ($\vec{\tau}$): $\vec{\tau} = \vec{p} \times \vec{E} = pE \sin\theta$
    • Potential Energy ($U$): $U = -\vec{p} \cdot \vec{E} = -pE \cos\theta$
    • Stable Equilibrium: $\theta = 0^\circ$
    • Unstable Equilibrium: $\theta = 180^\circ$

• Electric Flux ($\Phi_E$)

  • Definition: Measure of total electric field lines passing through a surface
  • Formula: $\Phi_E = \vec{E} \cdot \vec{A} = EA \cos\theta$
  • Unit & Type: $\text{N}\cdot\text{m}^2/\text{C}$ (Scalar quantity)

• Gauss's Law

  • Statement: Total flux through any closed surface equals $\frac{1}{\varepsilon_0}$ times net enclosed charge
  • Formula: $\Phi_E = \oint \vec{E} \cdot d\vec{A} = \frac{q_{\text{enclosed}}}{\varepsilon_0}$
  • Gaussian Surface: Imaginary symmetric closed surface surrounding charge
  • Applications
    • Infinitely Long Wire: $E = \frac{\lambda}{2\pi\varepsilon_0 r}$ ($\lambda$ = linear charge density)
    • Infinite Plane Sheet: $E = \frac{\sigma}{2\varepsilon_0}$ ($\sigma$ = surface charge density)
    • Thin Uniform Spherical Shell
      • Outside Shell ($r > R$): $E = \frac{1}{4\pi\varepsilon_0} \frac{q}{r^2}$
      • On Surface ($r = R$): $E = \frac{1}{4\pi\varepsilon_0} \frac{q}{R^2}$
      • Inside Shell ($r < R$): $E = 0$