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Electrostatic Force and Electrostatic field:-

Electrostatic:- It is a branch of physics that deals with the phenomena and properties of stationary or slow-moving electric charges with no acceleration.
Coulomb’s Law:- It states that the electro-static force of attraction or repulsion between two charged bodies is directly proportional to the product of their charges and varies inversely as the square of the distance between the two bodies

F = Kq₁q₂/r²Here, K = 1/4πε₀ = 9×10⁹ Nm²C^-2 (in free space)

Relative Permittivity (ε_r):- The relative permittivity (ε_r) of a medium is defined as the ratio between its permittivity of the medium (ε) and the permittivity (ε₀) of the free space.

ε_r = ε/ε₀

Coulomb force in vector form:- The force on charge q₁ due to q₂ is,

       If q₁q₂>0, R.H.S is positive.

If q₁q₂<0, a negative sign from q₁q₂ will change  and .  The relation will again be true, since, in that case  have same directions.

Unit of Charge:-

       C.G.S, q = ±1 stat-coulomb

       S.I, q = ±1 Coulomb

Relation between coulomb and stat-coulomb:-

        1 coulomb = 3×10⁹ stat-coulomb

        1 coulomb =(1/10) ab-coulomb (e.m.u of charge)

Dielectric constant:- The dielectric constant (ε_r) of a medium can be defined as the ratio of the force between two charges separated by some distance apart in free space to the force between the same two charges separated by the same distance apart in that medium.

          So, ε_r = ε/ε₀ = F₁/F₂

Here, F₁ and F₂ are the magnitudes of the force between them in free space and in a medium respectively.

Charges:-

?         Line charge, λ = q/L

        Surface charge, σ = q/A

        Volume charge, ρ = q/V

Electric field ( $\vec{E}$ ) :- The strength of an electric field is measured by the force experienced by a unit positive charge placed at that point. The direction of field is given by the direction of motion of a unit positive charge if it were free to move.

Unit of Electric field:-

         E = [Newton/Coulomb] or [Joule/(Coulomb) (meter)]

Electric lines of force:- An electric line of force is defined as the path, straight or curved, along which a unit positive charge is urged to move when free to do so in an electric field. The direction of motion of unit positive charge gives the direction of line of force.

     Properties:-

 (a) The lines of force are directed away 
from a positively charged conductor and are directed towards a 
negatively charged conductor.

  (b) A line of force starts from a 
positive charge and ends on a negative charge. This signifies line of 
force starts from higher potential and ends on lower potential.

Electric field intensity due to a point charge:- E = (1/4πε₀) (q/r²)
Electric field Intensity due to a linear distribution of charge:-

         (a) At point on its axis.

         E = (λ/4πε₀) [1/a – 1/a+L]

         Here, λ is the linear charge density.

        (b) At a point on the line perpendicular to one end.

         Here λ is the line charge.

Electric field due to ring of uniform charge distribution:-

        At a point on its axis, E = (1/4πε₀) [qx/(a²+x²)^3/2]

Electric field due to uniformly charged disc:-

        Here σ is the surface charge.

Electric field due to thin spherical shell:-

         (a) E_out = (1/4πε₀) (q/r²)

          (b) E_in = 0

Electric field of a non-conducting solid sphere having uniform volume distribution of charge:-

         (a) Outside Point:- E_out = (1/4πε₀) (Q/r²)

         (b) Inside Point:- E_in = (1/4πε₀) (Qr/R³)

         (c) On the Surface:- E_surface = (1/4πε₀) (Q/R²)

        Here, Q is the total charge

Electric field of a cylindrical conductor of infinite length having line charge λ:-

      (a) Outside the cylinder:- E = λ/2πε0r

      (b) Inside the cylinder:- E = 0

Electric field of a non-conducting cylinder having uniform volume density of charge:-

       (a) Outside the cylinder:- E = λ/2πε₀r

       (b) Inside a point:- E = ρr/2ε₀

Electric field of an infinite plane sheet of charge surface charge (σ) :- E = σ/2ε₀

Electric field due to two oppositely infinite charged sheets:-

          (a) Electric field at points outside the charged sheets:-

          E_P= E_R= 0

         (b) Electric field at point in between the charged sheets:-

         E_Q = σ/ε₀

Electric Dipole:- An electric dipole consists of two equal and opposite charges situated very close to each other.

Dipole Moment:- Dipole moment ( $\vec{P}$ ) of an electric dipole is defined as the product of the magnitude of one of the charges and the vector distance from negative to positive charge.

Unit of Dipole Moment:- coulomb meter (S.I), stat coulomb cm (non S.I)

Electric field due to an electric dipole:-

        (a) At any point on the axial line:-

Alt Tag: Electric field due to an electric dipole on the axial line.

        (b) At a point on the equatorial line (perpendicular bisector):-

        (c) At any point:-

Torque ( $\tau$ ) acting on a electric dipole in a uniform electric field (E):-

          = pE sinθ

       Here, p is the dipole moment and θ is the angle between direction of dipole moment and electric field E.

Electric Flux:- Electric flux ?_E for a surface placed in an electric field is the sum of dot product of $\vec{E}$ and $d\vec{a}$ for all the elementary areas constituting the surface.

Gauss Theorem:- It states that, for any distribution of charges, the total electric flux linked with a closed surface is 1/ε₀ times the total charge with in the surface.

Electric field (E) of an infinite rod at a distance (r) from the line having linear charge density (λ):-

         E = λ/2πε₀r

       The direction of electric field E is radially outward for a line of positive charge.

Electric field of a spherically symmetric distribution of charge of Radius R:-

     (a) Point at outside (r > R):- E = (1/4πε₀) (q/r²), Here q is the total charge.

    (b) Point at inside (r < R):- E = (1/4πε₀) (qr/R³), Here q is the total charge.

Electric field due to an infinite non-conducting flat sheet having charge σ:-

        E = σ/2ε₀

        This
 signifies, the electric field near a charged sheet is independent of 
the distance of the point from the sheet and depends only upon its 
charge density and is directed normally to the sheet.

Electric field due to an infinite flat conductor carrying charge:-

          E= σ/ε₀

Electric pressure (P_elec) on a charged conductor:-

         P_elec = (½ε₀) σ²


 

   
Electro-Static Potential and Capacitance

Electric Potential:-

              (a) Electric potential, at any point, is defined as the
 negative line integral of electric field from infinity to that point 
along any path.

           (b) V(r) = kq/r

          (c) 
 Potential difference, between any two points, in an electric field is 
defined as the work done in taking a unit positive charge from one point
 to the other against the electric field.

W_AB = q [V_A-V_B]

So, V = [V_A-V_B] = W/q

Units:- volt (S.I), stat-volt (C.G.S)

Dimension:- [V] = [ML²T^-3A^-1]

Relation between volt and stat-volt:- 1 volt = (1/300) stat-volt

Relation between electric field (E) and electric potential (V):-

         E = -dV/dx = --dV/dr

Potential due to a point charge:-

        V = (1/4π ε₀) (q/r)

Potential at point due to several charges:-

         V = (1/4π ε₀) [q₁/r₁ + q₂/r₂ + q₃/r₃]

           = V₁+V₂+ V₂+….

Potential due to charged spherical shell:-

         (a) Outside, V_out = (1/4π ε₀) (q/r)

        (b) Inside, V_in = - (1/4π ε₀) (q/R)

        (c) On the surface, V_surface = (1/4π ε₀) (q/R)

Potential due to a uniformly charged non-conducting sphere:-

        (a) Outside, V_out = (1/4π ε₀) (q/r)

        (b) Inside, V_in = (1/4π ε₀) [q(3R²-r²)/2R³]

        (c) On the surface, V_surface = (1/4π ε₀) (q/R)

       (d) In center, V_center = (3/2) [(1/4π ε₀) (q/R)] = 3/2 [V_surface]

Common potential (two spheres joined by thin wire):-

       (a) Common potential, V = (1/4π ε₀) [(Q₁+Q₂)/(r₁+r₂)]

       (b) q₁ = r₁(Q₁+Q₂)/(r₁+r₂) = r₁Q/ r₁+r₂ ;  q₂ =  r₂Q/ r₁+r₂

       (c) q₁/q₂ = r₁/r₂ or σ₁/ σ₂ = r₁/r₂

Potential at any point due to an electric dipole:-

       V (r,θ) = qa cosθ/4πε₀r² = p cosθ/4πε₀r²

     (a) Point lying on the axial line:- V = p/4πε₀r²

     (b) Point situated on equatorial lines:- V = 0

If n drops coalesce to form one drop, then,

      (a) R = n^1/3r

     (b) Q = nq

     (c) V = n^2/3V_small

    (d) σ = n^1/3 σ_small

    (e) E = n^1/3 E_small

Electric potential energy U or work done of the system W having charge q₁ and q₂:-

     W = U = (1/4πε₀) (q₁q₂/r₁₂) = q₁V₁

Electric potential energy U or work done of the system W of a three particle system having charge q₁, q₂ and q₃:-

     W = U = (1/4πε₀) (q₁q₂/r₁₂ + q₁q₃/r₁₃ + q₂q₃/r₂₃)

Electric potential energy of an electric dipole in an electric field:- Potential energy of an electric dipole, in an electrostatic field, is defined as the work done in rotating the dipole from zero energy position to the desired position in the electric field.

      (a) If θ = 90º, then W = 0

      (b) If θ = 0º, then W = -pE

      (c) If θ = 180º, then W = pE

Kinetic energy of a charged particle moving through a potential difference:-

       K. E = ½ mv²= eV

Conductors:- Conductors are those substance through which electric charge easily.
Insulators:- Insulators (also called dielectrics) are those substances through which electric charge cannot pass easily.
Capacity:- The capacity of a conductor is defined as the ratio between the charge of the conductor to its potential

       C = Q/V

      Units:-

     S.I – farad (coulomb/volt)

     C.G.S – stat farad (stat-coulomb/stat-volt)

    Dimension of C:- [M^-1L^-2T⁴A²]

Capacity of an isolated spherical conductor:-

       C = 4πε₀r

Capacitor:- A capacitor or a condenser is an arrangement which provides a larger capacity in a smaller space.
Capacity of a parallel plate capacitor:-

        C_air= ε₀A/d

       C_med = Kε₀A/d

       Here, A is the common area of the two plates and d is the distance between the plates.

Effect of dielectric on the capacitance of a capacitor:-

        C = ε₀A/[d-t+(t/K)]

        Here d is the separation between the plates, t is the thickness of the dielectric slab A is the area and K is the dielectric constant of the material of the slab.

        If the space is completely filled with dielectric medium (t=d), then,

       C = ε₀KA/ d

Capacitance of a sphere:-

       (a) C_air = 4πε₀R

      (b) C_med = K (4πε₀R)

Capacity of a spherical condenser:-

      (a) When outer sphere is earthed:-

      C_air = 4πε₀ [ab/(b-a)]

     C_med = 4πε₀ [Kab/(b-a)]

     (b) When the inner sphere is earthed:-

     C₁= 4πε₀ [ab/(b-a)]

     C₂ = 4πε₀b?

     Net Capacity, C '=4πε₀[b²/b-a]

    Increase in capacity, ΔC = 4π ε₀b

    It signifies, by connecting the inner 
sphere to earth and charging the outer one we get an additional capacity
 equal to the capacity of outer sphere.

Capacity of a cylindrical condenser:-

       C_air = λl / [(λ/2π ε₀) (log_eb/a)] = [2π ε₀l /(log_eb/a) ]

       C_med = [2πKε₀l /(log_eb/a) ]

Potential energy of a charged capacitor (Energy stored in a capacitor):-

      W = ½ QV = ½ Q²/C = ½ CV²

Energy density of a capacitor:-

      U = ½ ε₀E² = ½ (σ²/ ε₀)

This 
signifies the energy density of a capacitor is independent of the area 
of plates of distance between them so long the value of E does not change.

Grouping of Capacitors:-

         ?(a)

         (i) Capacitors in parallel:- C = C₁+C₂+C₃+…..+C_n

The resultant capacity of a number of capacitors, connected in parallel, is equal to the sum of their individual capacities.

       (ii)V₁= V₂= V₃ = V

       (iii) q₁ =C₁V, q₂ = C₂V,  q₃ = C₃V

      (iv) Energy Stored, U = U₁+U₂+U₃

(b)

      (i) Capacitors in Series:- 1/C = 1/C₁+ 1/ C₂+……+ 1/C_n

?The
 reciprocal of the resultant capacity of a number of capacitors, 
connected in series, is equal to the sum of the reciprocals of their 
individual capacities.

         (ii) q₁ = q₂ = q₃ = q

        (iii) V₁= q/C₁, V₂= q/C₂, V₃= q/C₃

        (iv) Energy Stored, U = U₁+U₂+U₃

Energy stored in a group of capacitors:-

      (a) Energy stored in a series combination of capacitors:-

       W = ½ (q²/C₁) + ½ (q²/C₂) + ½ (q²/C₃) = W₁+W₂+W₃

      Thus, net energy stored in the combination is equal to the sum of the energies stored in the component capacitors.

      (b) Energy stored in a parallel combination of capacitors:-

       W = ½ C₁V ² +½ C₂V ² + ½ C₃V ² = W₁+W₂+W₃

      The net energy stored in the combination is equal to sum of energies stored in the component capacitors.

Force of attraction between plates of a charged capacitor:-

      (a) F = ½ ε₀E²A

      (b) F = σ²A/2ε₀

      (c) F=Q²/2ε₀A

Force on a dielectric in a capacitor:-

       F = (Q²/2C²) (dC/dx) = ½ V² (dC/dx)

Common potential when two capacitors are connected:-

        V = [C₁V₁+ C₂V₂] / [C₁+C₂] = [Q₁+Q₂]/ [C₁+C₂]

Charge transfer when two capacitors are connected:-

       ΔQ = [C₁C₂/C₁+C₂] [V₁-V₂]

Energy loss when two capacitors are connected:-

         ΔU = ½ [C₁C₂/C₁+C₂] [V₁-V₂] ²

Charging of a capacitor:-

         (a) Q = Q₀(1-e^-t/RC)

         (b) V = V₀(1-e^-t/RC)

         (c) I = I₀(1-e^-t/RC)

         (d) I₀ = V₀/R

Discharging of a capacitor:-

        (a) Q = Q₀(e^-t/RC)

        (b) V = V₀(e^-t/RC)

         (c)  I = I₀(e^-t/RC)

Time constant:- $\tau = RC$

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Electrostatic Force and Electrostatic field:-

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