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Electrostatics

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Electrostatics PDF Notes, Important Questions and Synopsis

SYNOPSIS

  • Electric charge: The property associated with matter due to which it produces and experiences electric and magnetic fields.
  • Properties of charge: Transferable, always associated with mass, conserved, quantised.
  • Methods of charging of a body: Conduction, induction, friction

  • Coulomb’s law:
     begin mathsize 12px style straight F with rightwards arrow on top equals fraction numerator 1 over denominator 4 πε subscript 0 straight epsilon subscript straight r end fraction fraction numerator straight q subscript 1 straight q subscript 2 over denominator open vertical bar straight r with rightwards arrow on top close vertical bar squared end fraction straight r with hat on top end style
    Not valid for distance less than .10-2
    Electrostatic force is conservative in nature.
    Valid only for point charges.
  • Superposition principle: The total force on a given charge is the vector sum of all the individual forces exerted by each of the other charges.
  • Electric field: The space around the charge in which its influence can be felt by any other charged particle.
    A point charge in an electric field experiences force .
    begin mathsize 12px style straight F with rightwards arrow on top equals straight q subscript 0 straight E with rightwards arrow on top end style

    For positive source charge, the electric field is radially outward, whereas for negative source charge, the electric field is radially inward.
    A positive charge placed in an electric field experiences force in the same direction as begin mathsize 12px style top enclose straight E end style , while a negative charge experiences force in the opposite direction as begin mathsize 12px style top enclose straight E end style.
  • Properties of lines of force: The strength of the electric field is directly proportional to the density of lines of force.
    Electric field within a conductor is zero.
  • Electric dipole:
    It is a system of two equal and opposite point charges separated by a very small and finite distance. The strength of an electric dipole is measured by electric dipole moment p = 2ql; the direction is from –q to +q.
    Electric field due to a dipole at some general point is given by
    begin mathsize 12px style straight E with rightwards arrow on top equals fraction numerator straight p over denominator 4 πε subscript 0 straight r cubed end fraction square root of 1 plus 3 cos squared straight theta end root end style, where is the angle between the axis of the dipole and the line joining point P and the axis.
  • Dipole in uniform electric field:
    Net force on a dipole in a uniform electric field is zero.
    Torque begin mathsize 12px style straight tau equals pEsinϕ end style , torque is maximum when the dipole is perpendicular begin mathsize 12px style open parentheses straight ϕ equals 90 to the power of 0 close parentheses end style to the field and minimum when the dipole is parallel begin mathsize 12px style open parentheses straight ϕ equals   0 to the power of 0 close parentheses end style or antiparallel begin mathsize 12px style open parentheses straight ϕ equals 180 to the power of 0 close parentheses end style to the field. 

  • Dipole in non-uniform electric field:
    The net force depends on 
    Orientation of the dipole, dipole moment of the dipole and how rapidly the field varies. 

  • Electric flux: 
    The product of magnitude of electric field begin mathsize 12px style top enclose straight E end style and surface area A perpendicular to the field is called electric flux begin mathsize 12px style straight ϕ subscript straight E end style . begin mathsize 12px style straight ϕ subscript straight E end style depends on both field pattern and surface. The net flux is directly proportional to the net number of lines leaving the surface.

  • Gauss’s law:
    Electric flux through a closed surface S is given by begin mathsize 12px style straight q over straight epsilon subscript 0 end style , where q is the charge enclosed by surface S.
    The net field begin mathsize 12px style top enclose straight E end style is due to all charges present inside and outside the Gaussian surface. No net flux is contributed by the charges present outside the Gaussian surface because the number of E-lines coming into the surface is equal to the number of E-lines going out of the surface.

  • Electric field due to a charged isolated plate:
    The electric field due to a charged isolated plate is twice the field due to a plane sheet of charge . begin mathsize 12px style straight E equals fraction numerator straight sigma over denominator straight epsilon blank presubscript 0 end fraction end style The electric field due to an infinitely charged plate does not depend on the distance from the plate.

  • Electric potential:
    The potential at a point P is given by begin mathsize 12px style straight V equals fraction numerator straight W subscript ext open parentheses straight infinity rightwards arrow straight P close parentheses over denominator straight q end fraction equals negative integral from straight infinity to straight P of straight E with rightwards arrow on top dl end style, which is equal to the external work done per unit positive charge in shifting slowly from infinity to this point. 
    In the direction of electric field, the potential always decreases.
    Potential is a scalar quantity. Its SI unit is volt.

     begin mathsize 12px style text W end text subscript ext open parentheses straight A rightwards arrow straight B close parentheses end subscript equals straight V subscript straight B minus straight V subscript straight A equals negative integral from straight A to straight B of straight E with rightwards arrow on top dl with rightwards arrow on top end style
    begin mathsize 12px style table attributes columnalign left end attributes row cell straight W subscript ext open parentheses straight A rightwards arrow straight B close parentheses end subscript equals straight V subscript straight B minus straight V subscript straight A equals negative integral from straight A to straight B of straight E with rightwards arrow on top dl with rightwards arrow on top end cell row cell straight W subscript ext greater than 0 comma straight V subscript straight B greater than straight V subscript straight A end cell row cell straight W subscript ext equals 0 comma straight V subscript straight B equals straight V subscript straight A end cell row cell straight W subscript ext less than 0 comma straight V subscript straight B less than straight V subscript straight A end cell end table end style

  • Equipotential surface:
    An equipotential surface is a surface with constant value of potential at all points on the surface. The equipotential surface of a single point charge are concentric spherical surfaces centred at the charge. For uniform electric field, equipotential surfaces are planes normal to the plane of the electric field.

  • Potential energy of a system of charges:

    For two charges, 
    begin mathsize 12px style straight U equals fraction numerator 1 over denominator 4 πε subscript 0 end fraction fraction numerator straight q subscript 1 straight q subscript 2 over denominator straight x end fraction end style  

    For three charges,
     begin mathsize 12px style straight U equals fraction numerator 1 over denominator 4 πε subscript 0 end fraction open square brackets fraction numerator straight q subscript 1 straight q subscript 2 over denominator straight x subscript 1 end fraction plus fraction numerator straight q subscript 2 straight q subscript 3 over denominator straight x subscript 2 end fraction plus fraction numerator straight q subscript 3 straight q subscript 1 over denominator straight x subscript 3 end fraction close square brackets end style 

    For discreet system of charges,  
    begin mathsize 12px style straight U equals 1 half fraction numerator 1 over denominator 4 πε subscript 0 end fraction sum for straight i not equal to straight j of fraction numerator straight q subscript straight i straight q subscript straight j over denominator straight r subscript ij end fraction end style

    General formula for calculating potential energy: U = qV, where V is the potential at a point of some continuous body (ring, disc, solid sphere etc.).

  • Finding electric field from electric potential:

    begin mathsize 12px style table attributes columnalign left end attributes row cell straight E subscript straight s equals negative open parentheses dV over ds close parentheses end cell row cell straight E subscript straight x equals negative fraction numerator partial differential straight V over denominator partial differential straight x end fraction semicolon straight E subscript straight y equals negative fraction numerator partial differential straight V over denominator partial differential straight y end fraction semicolon straight E subscript straight z equals negative fraction numerator partial differential straight V over denominator partial differential straight z end fraction end cell end table end style

  • Earthing of a conductor:
    On earthing a conductor, the charge on the conductor varies and the potential of the conductor becomes zero.

  • Charge distribution on a conductor surface:
    The charge distribution takes place on the surface of the conductor in such a way that the product of charge density begin mathsize 12px style straight sigma end style  and the radius of curvature r at any point on the conductor is constant, i.e.begin mathsize 12px style σr equals straight c end style  

  • Dielectrics are non-conducting substances. They have no charge carriers.
  • The molecules of a substance may be polar or non-polar.
  • In a non-polar molecule, the centres of positive and negative charges coincide. Thus, the molecule does not have any permanent dipole moment.
    In a polar molecule, the centres of positive and negative charges are separated. These molecules have a permanent dipole moment.
    Introduction of dielectrics develops a net dipole moment in the substance whether polar or non-polar.
  • Capacitance is determined purely geometrically, by the shapes, sizes and relative positions of the two conductors.
  • Changes observed when the medium between the plates of a capacitor is filled with an insulating substance (dielectric):
  1. Polarization of the medium gives rise to a field in the opposite direction. The net electric field inside the insulating medium is reduced.
  2. Potential difference between the plates is thus reduced. 
  3. Capacitance C increases from its value C0 when there is no medium (vacuum).
    C = KC
    Where K is the dielectric constant of the insulating substance.
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