Electrostatics II

Gauss’s law in electrostatic
Electrostatic potential
Electric Dipole
Capacitors
Grouping of Capacitors




Flux of An Electric Field Or Electric Flux

Electric flux is defined as the number of field lines that pass through a given surface . In Figure, lines of electric flux emerging from a point charge pass through an imaginary spherical surface with the charge at its center.

This definition can be expressed as follows: Φ = ∑E · A , where Φ (the Greek letter phi) is the electric flux, E is the electric field, and A is area perpendicular to the field lines.
Electric flux is measured in N · m2 / C2 and is a scalar quantity. If the surface under consideration is not perpendicular to the field lines, then the expression is Φ = ∑ EA cos θ .

In general terms, flux is the closed integral of the dot product of the electric field vector and the vector ΔA . The direction of ΔA is the outward drawn normal to the imaginary surface.
Mathematically, Φ = ΦE · dA . The accepted convention is that flux lines are positive if leaving a surface and negative if entering a surface.

Gauss’s law in electrostatics or Gauss’s theorem







This law gives a relation between the electric flux through any closed hypothetical surface (called Gussian surface) and the charge enclosed by any surface. It states, " The electric flux through any closed surface is equal to     times the 'net' charge enclosed by the surface."



That is,

Application of Gauss’s law









Gauss's law is useful when there is symmetry in the charge distribution, as in case of uniformly charged sphere, long cylinders, and flat sheets. In such cases, it is possible to find a simple Gaussian surface over which the surface integral given by Φ = ∑E · A can be easily evaluated.

Electric Field Due To A Charged Spherical Shell












Figure (a) Charged spherical shell of radius R. (b) A gaussian sphere with radius r > R.
(c) A gaussian sphere with radius r < R.


When outside the shell of charge, as in Figure 8 (a), the left side of Gauss's equation reduces to the following expression:




Thus, the electric field outside a charged sphere is the same as if the same amount of charge were concentrated in a point located at the center of the sphere.

The gaussian surface inside the sphere encloses no charge, and therefore, there is no electric field inside the uniformly charged spherical shell . The same proof holds within a solid conductor because all the charge of the conductor resides on the surface. Because the electric field inside even an irregularly shaped conductor is zero, the charge will not be evenly distributed over an irregular shape. The charge will tend to accumulate on protruding points on the outside of the conductor.
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Electrostatic potential

Imagine moving a small test charge q′ from point A to point B in the uniform field between parallel plates. The work done in transferring the charge equals to the product of the force on the test charge and the parallel component of displacement.

This work can also be expressed in terms of E from the definition of electric field as the ratio of force to charge: W · d, E = F/ q and W = q′ . See Figure .

Work is change in potential energy: U B − U A = q′ Ed .

In general, the electrostatic potential difference, sometimes called the electric potential difference , is defined as the energy change per unit positive charge, or V B − V A = ( U B − U A )/ q′ .

For certain configurations of electric field, it may be necessary to use the integral definition of electrostatic potential:




where a test charge moves over a line integral from point A to point B along path s in an electric field (E).

For the special case of Parallel Plates :





where V is the potential difference between the plates, measured in units of volts (V):




The electric potential due to a point charge (q) at a distance (r) from the point charge is




Figure
Work is done when q′ moves from position
A to B in an electric field E.












Equipotential Surfaces

Equipotential surfaces are surfaces where no work is required to move a charge from one point to another . The equipotential surfaces are always perpendicular to the electric field lines .

Equipotential lines are two-dimensional representations of the intersection of the surface with the plane of the diagram. In Figure , equipotential lines are shown for (a) a uniform field, (b) a point charge, and (c) two opposite charges.










Figure Equipotential lines for (a) a uniform electric field, (b) a point charge, and (c) two opposite charges.


The Electrical Potential Energy of a pair of point charges separated by a distance r is




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Electric Dipole

Two equal and opposite point charges placed at a short distance apart constitute an electric dipole.



Electric Dipole Moment


Electric dipole moment is a vector directed     along the axis of the dipole, from the negative to the positive charge.

The magnitude of the dipole moment is



where 2a is the distance between the two charges.



Electric Dipole In A Uniform Electric Field


If the electric field strength E at every point in the field is the same, then it is said to be a uniform electric field . Consider an electric dipole consisting of two equal and opposite point charges +q and –q separated by a distance 2a.














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Capacitor



Capacitor is an arrangement of two conductors carrying charges of equal magnitude and opposite sign and seperated by an insulating medium.

The following points may be carefully noted:

(i) The net charge on the capacitor as a whole is zero.

(ii) The positively charged conductor is at a higher potential than negatively charged conductor. The potential difference V between the conductors is proportional to the magnitude of charge Q and the ratio Q/V is known as capacitance C of the capacitor.







(iii) In a circuit, a capacitor is represented by the symbol:








Parallel Plate Capacitor


Consider a parallel plate capacitor consisting of two parallel plates of area A square metres separated by a distance d as shown in the figure.








Energy Stored In Charged Capacitor Capacitor






When the capacitor store charge, capacitors are also storing energy:

Energy, E = 1/2 QV = 1/2 CV2 where E = energy in joules (J).

Note that capacitors return their stored energy to the circuit . They do not 'use up' electrical energy by converting it to heat as a resistor does. The energy stored by a capacitor is much smaller than the energy stored by a battery so they cannot be used as a practical source of energy for most purposes.

Parallel Plate Capacitor with Dielectric





The capacitance of a set of charged parallel plates is increased by the insertion of a dielectric material. The capacitance is inversely proportional to the electric field between the plates , and the presence of the dielectric reduces the effective electric field. The dielectric is characterized by a dielectric constant k, and the capacitance is multiplied by that factor.


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Grouping of Capacitors


Replacing a combination of capacitors by a single equivalent capacitor is called
'grouping of capacitors '. It simplifies the problem and is divided into two types:

(i) Series Combination of of Capacitors .
(ii) Parallel Combination of of Capacitors .






Difference Between Gravitational Force And Electrostatic Force





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