In this lecture we started Electrostatics. The basic question of electrostatics is to find the force on a test charge at some (field) due to a source charge or a system of source charges fixed at some (source) point or points.
Superposition principle for the force simplifies ths to finding the force on the test charge due to each of the source charges and doing a vector sum for the total force.
The force between two charges is given by the fundamental law of electrostatics which is the Coulombs Law. We can then define the Electric Field as a vector field E whose magnitude is the force on a unit test charge at that point. Field then maybe expressed as a vector function of the co-ordinates in a region such that any test charge feels a force when placed in that region.
The electric field due to a point charge can easily be calculated from coulombs law. The expression for the field due to a point charge can then be generalized for a system of discrete charges through superposition by summing over the individual electric fields due to each charge and finally to a continuous distribution through an integral over the charge distribution. Note that there can be 3 types of charge distributions; line , surface and volume characterized by the respective densities which are in general functions of the source co-ordinates eg. ρ(r prime) Most often in problems we will deal with constant or uniform charge distributions although in certain cases we may use a charge distribution which is not uniform.
This reduces the determination of the electric field ( force on unit test charge) at some point to solving an integral with a vector integrand. This is not simple to perform in most cases except a few. So we need to seek alternate ways by studying the properties of the electric field.
It is easily seen in the field line picture that the FLUX of the electric field over a closed surface S is a measure of the charge enclosed within this surface. For a single point charge the flux is given by (q/epsilon zero). This result is easily generalized to a discrete system or a continuous distribution of charge by the superposition principle to give the flux of E as
∫ S E.n da=Q/ε 0 where Q is the total charge enclosed by S. Using the divergence theorem the surface maybe related to a volume integral over a volume V which is enclosed by the closed surface S of the div E. Comparison of two sides after writing the total charge Q also as a volume integral of a volume charge density ρ over V now provide the relation that'
∇ .E= ρ / ε 0
These both are the Integral and Differential form of Gauss Law respectively.
Note that Gauss Law is just another expression for Coulombs Law which is the fundamental law. The Gauss law is a consequence of the Coulombs Law.
sir, can u please comment on dis, dt i, j and k are constant while r, theta and fi r nt(in plane polar)??
ReplyDeleteThe Cartesian Unit Vectors i, j, k are fixed in direction once you have chosen your co-ordinate axes. But in curvilinear co-ordinates the unit vectors change direction from one point to another.
ReplyDeleteRemember the direction of unit vectors are always in the direction of the increasing coordinate, but in curvilinear coordinates some of the coordinates are angles like theta and phi so their increasing direction is always tangential to some curve and this tangent vector will point in different directions at different point of the curve which describes the increase of phi. Contrast this with x, y ,z which always increases
along a straight line and the tangent to a straight line is the straight line itself, it never changes direction so i j k never chnage directions onec a coordinate axes has been chosen.
ok sir, so change in these vectors is due to the continous change in direction as we go. as far as magnitude is concerned, things are exactly the same as wwe can see, magnitude is 1 for all of them. right?
ReplyDeleteSince they are unit vectors their magnitude has to be 1. In practice first the direction is obtained from the tangent vectors and then the tangent vectors are divided by their norm to obtain the unit ( tangent) vectors.
ReplyDeleteHowever even though magnitude remains the same because their direction changes their derivatives
exist and they cannot be taken outside integral signs. So great care has to be taken in integrating and differentiating these vectors.