Electrostatic Potential And Capacitance

- A quick revision of all the important concepts
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ELECTROSTATIC POTENTIAL

Electric Potential energy
  • Work done by external forces in moving a charge q from R to P is
  • The electrostatic potential difference between two points in an electric field is defined as the amount of work done in moving a unit positive test charge from one point to the other point against electrostatic force without any acceleration.
  • This is a conservative force and hence depends only on the initial and final positions of the charge.
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Potential Energy
Potential energy of charge q at a point is the work done by the external force in bringing the charge q from infinity to that point.

Electrostatic Potential
  • The electrostatic potential at any point in an electric field is equal to the amount of work done per unit positive test charge or in bringing the unit positive test charge from infinity to that point, against the electrostatic force without acceleration.

SI unit is volt (V) and 1V = 1J/C.
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Potential due to a point charge
  • The potential at P due to the charge Q is


Potential due to an Electric dipole
  • The electric potential of a dipole is given by
  • Potential on the dipole axis is given by

Positive sign for negative sign for .
The potential in the equatorial plane is zero.
  • The potential due to a dipole depends on the angle between the position vector r and the dipole moment vector p.
  • The electric dipole potential falls off, at large distance, asnot as
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Potential Due To A System Of Charges
Consider a system of charges q1, q2,..., qn with position vectors r1, r2,...,rn.The total charge configuration is the algebraic sum of the potentials due to the individual charges



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Equipotential Surfaces
  • An equipotential surface is a surface with a constant value of potential at all points on the surface.
  • The shape of equipotential surface due to
    • line charge is cylindrical.
    • point charge is spherical
  • Equipotential surfaces do not intersect each other as it gives two directions of electric field E at intersecting points which is not possible.
  • Electric field is always normal to an equipotential surface at every point of it and directed from one equipotential surface at higher potential to the equipotential surface at lower potential.
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Relation between E and V
  • Electric field is in the direction in which the potential decreases steepest.
  • Its magnitude is given by the change in the magnitude of potential per unit displacement normal to the equipotential surface at the point.
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Potential Energy Of A System Of Charges
  • The potential energy of a system of two charges q1 and q2 is
  • If q1 q2 > 0potential energy is positive because for like charges, electrostatic force is repulsive and a positive amount of work is needed to be done against this force to bring the charges from infinity to a finite distance apart.
  • For unlike charges(q1 q2 < 0), the electrostatic force is attractive. In that case, a positive amount of work is needed against this force to take the charges from the given location to infinity,so the potential energy is negative.
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Potential Energy In An External Field
  • Potential energy of a single charge is given by

whereV(r) is the external potential at the point r. The unit of energy is defined as 1 electron volt
  • Potential energy of a system of two charges in an external field is given by
  • Potential energy of a dipole in an external field is given by
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Electrostatics Of Conductors
  • The electrostatic field inside a conductor is zero
  • Electrostatic field at the surface of a charged conductor must be normal to the surface at every point.
  • There is no net charge at any point inside the conductor, and any excess charge must reside at the surface.
  • Electrostatic potential is constant throughout the volume of the conductor and has the same value (as inside) on its surface
  • Electric field at the surface of a charged conductor is
  • The process which involves the making of a region free from any electric field is known as electrostatic shielding.
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Dielectrics
Dielectrics are non-conducting substances.
  • Non-polar molecules:Centres of positive andnegative charges coincide and have no permanent dipole moment. Eg:
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  • Polar molecules: centres of positive and negative charges are separated and have permanent dipole moments. Eg:
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Polarisation
The dipole moment per unit volume is called polarisation and is denoted by P.

Where
is a constant characteristic of the dielectric and is known as the

electric susceptibility of the dielectric medium.

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CAPACITANCE

Capacitors
  • A capacitor is a system of two conductors separated by an insulator.
  • The conductors may be so charged by connecting them to the two terminals of a battery.
  • The total charge of the capacitor is zero.
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Capacitance
  • The constant C is called the capacitance of the capacitor.
  • The capacitance C depends only on the geometrical configuration of the system of two conductors.
  • The SI unit of capacitance is 1 farad (=1 coulomb volt-1)
Parallel plate capacitor
  • A parallel plate capacitor consists of two large plane parallel conducting plates separated by a small distance.
  • The capacitance C of the parallel plate capacitor is
  • The product is called the permittivity of the medium and is denoted by

For vacuum K = 1,: is called the permittivity of vacuum. Kis called the dielectric constant of the substance.
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Combination Of Capacitors
  • Capacitors in series
The general formula for effective capacitance of a series combination of n capacitors is

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  • Capacitors in parallel
The general formula for effective capacitance C forparallel combination of n capacitors
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  • The energyU stored in a capacitor of capacitance C,with charge Q and voltage V is
  • The electric energy density (energy per unit volume) in a region with electric field is