Question

Electrostatics all formulae

Answer 1

Electrostatics formulasElectrostatic forceCoulomb's Law

F = kq1q2/r2

where k=1/4πεo=9x109Nm2C-2

εo = 8.85x10-12C2m-2N-1

Vector notation

Electrostatic fieldElectric field due to a point charge

E = F/qo = kq/r2 N/C

E due to circular loop of charge (radius r) at a distance x from the center

Electric dipoleDipole moment

 Cm

Electric field at an axial point of a dipole

Electric field at an equatorial point of a dipole

Torque acting on a dipole in a uniform electric field

Potential energy of a dipole in a uniform electric field

Electrostatic potentialElectrostatic potential difference

Potential due to a point charge

Potential at an axial point of a dipole

 if  then 

Potential at an equatorial point of a dipole

Relation between electrostatic field and potential gradientElectric field = negative of the potential gradient

Electrostatic potential energyElectrostatic potential energy of two point charges

Gauss' theoremElectric flux

Gauss' theorem

Definition: Electric flux ϕ through any closed surface is 1/εo times the net charge Q enclosed by the surface.

Electric field E due to infinitely long straight wire (a line charge)

Electric field E due to thin infinite plane sheet of charge

Electric field between two thin infinite plane parallel sheets of charge

Electric field due to uniformly charged spherical shell

 for r > R

 for r < R

 for r = R

Capacitance

 Farad 1F = 1 C/V

Isolated spherical conductor

Parallel plate capacitor

 or  where  and k is dielectric constant

Capacitors in series

Capacitors in parallel

Energy stored in a capacitor

Energy density 

Common potential

C with conducting slab between the two plates

 where t is thickness of slab [t < d]

C with dielectric slab between the two plates

 where k is the dielectric constant

527695e86d89c457be8719c13a192522.pdf

Answer 2

Formulas related to Electrostatics ie., static charges :

K = 1/(4πε) = 9 * 10⁹ units   ;;    ε = K * ε₀
Coulomb's law:  Force F = K Q1 Q2 / d² 

Electric field:  
  due to a point charge Q:  E = F / q  = K Q/d²
  a Long wire (infinite) of uniform charge density λ at a distance d:
      E =  K 2λ / d = λ / [2 πε d]
  due to a finite wire (length 2a) at distance d on the perpendicular bisector:
        E = K q /[x √(x²+a²) ]

  A large rectangular sheet (infinite): σ/(2ε)
  Inside a parallel plate capacitor (between two plates): σ/ε

Electric Flux: Φ = E * Area S : 
$\Phi=\int_S {\vec{E}.\vec{dS}}\\\\Gauss: \int_S {\vec{E}.\vec{dS}}=\frac{q_{enclosed}}{\epsilon_0}$

===============
Potential at point P
   V = - Work done in bringing a unit charge from infinity to point P
   V = W/ q
   E = -dV/dr 
$\vec{E}=-\frac{d\vec{E}}{d\vec{r}}$

Potential difference between points:
   $V=\int {\vec{E}.\vec{dr}}$

Potential :
    due to point charge: K q/r
    $V=K \int {\frac{dq}{r}}$
    across a parallel plate capacitor:  d * E = σ d /ε
    at a point close to a large sheet of charge = E d = σ d/2ε
    at the surface of a metallic sphere = K Q/R
    at a distance x from a ring of charge q ^ radius R, along the axis:
           V= Kq/√(R²+x²)
    Along the axis of a disc : V = σ/(2ε)  * [√(R²+x²) - x ]
    At the center of a disc:  V = σR / (2ε)
    At the edge of a disc:   V = σR /(π ε)

Potential energy 
    stored in a field E per unit volume:   U = 1/2 * ε * E²  
    stored in a system of two charges:  U = K q1 q2 / d
    work done = q V

==================
Capacitance = C = Q/V 
      parallel plate capacitor:  ε A / d  = σ A / (E d)
      A coaxial cable of inner and outer radii a & b, per unit length:  [2 K * Ln (a/b)]⁻¹
      Parallel combination: C = C1 + C2 + C3 + ...
      Series combination : 1/C = 1/C1 + 1/C2 + 1/C3 ...
      Of a sphere of radius R = R /K
      U = 1/2 * C V²

Dipole: charges along y axis
  Dipole moment = p = 2 a q
  Field E at point P (x,y,z): 
$E_x=-\frac{dV}{dx}=\frac{q}{4 \pi \epsilon_0} \{\frac{x}{[x^2+(y-a)^2+z^2]^{\frac{3}{2}}} - \frac{x}{[x^2+(y+a)^2+z^2]^{\frac{3}{2}}} \}\\\\E_y=-\frac{dV}{dy}=\frac{q}{4 \pi \epsilon_0} \{\frac{y-a}{[x^2+(y-a)^2+z^2]^{\frac{3}{2}}} - \frac{y+a}{[x^2+(y+a)^2+z^2]^{\frac{3}{2}}} \}\\\\E_z=-\frac{dV}{dz}=\frac{q}{4 \pi \epsilon_0} \{\frac{z}{[x^2+(y-a)^2+z^2]^{\frac{3}{2}}} - \frac{z}{[x^2+(y+a)^2+z^2]^{\frac{3}{2}}} \}$

Dipole:
     Along the axis at y: V = K P /(y² - a²)     ;;     E = K 2Py / (y² - a²)²
     Along the bisector:  V = 0     ;;  E = K P / (x² + a²)³/²

  Potential V at P(x,y,z) : 
$V=K \{\frac{q}{\sqrt{x^2+(y-a)^2+z^2}}-\frac{q}{\sqrt{x^2+(y+a)^2+z^2}}\}$

Field E due to a Ring of charge q and radius R at distance x along the axis:
   E = K q x /(x²+R²)³/² = K q Sin² θ * cosθ / R²

Torque on a dipole:
    $\vec{T}=\vec{P}X \vec{E}$
    Potential energy of a dipole in an electric field E:  U (θ) = - PE Cos θ = - P . E

Time period of oscillation of dipole in electric field E:
     T = 2 π √[ I /PE ]           I = moment of inertia of dipole

  Electric field inside a spherical shell = 0
  Electric field inside a conductor = 0
  Electric field inside a charged sphere:  K q r /R³
  Potential inside a charged sphere:   K q / R  *  [3/2 - r²/2R² ]