25-09-2014, 02:37 PM
ELECTRIC FIELD INTENSITY:
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If we consider one charge fixed in position, say Q1, and move a second charge
slowly around, we note that there exists everywhere a force on this second charge;
in other words, this second charge is displaying the existence of a force field that is
associated with charge, Q1. Call this second charge a test charge Qt . The force on it
is given by Coulomb’s law,
Writing this force as a force per unit charge gives the electric field intensity, E1 arising
from Q1:
E1 is interpreted as the vector force, arising from charge Q1 , that acts on a unit positive
test charge. More generally, we write the defining expression:
in which E, a vector function, is the electric field intensity evaluated at the test charge
location that arises from all other charges in the vicinity—meaning the electric field
arising from the test charge itself is not included in E.
ELECTRIC FLUX AND FLUX DENSITY:
About 1837 Michael Faraday performed an experiment.He had a pair of concentric
metallic spheres constructed, the outer one consisting of two hemispheres that could be
firmly clamped together. He also prepared shells of insulating material (or dielectric
material, or simply dielectric) that would occupy the entire volume between the
concentric spheres. He found that:
1. With the equipment dismantled, the inner sphere was given a known positive
charge.
2. The hemispheres were then clamped together around the charged sphere with
about 2 cm of dielectric material between them.
3. The outer sphere was discharged by connecting it momentarily to ground.4. The outer space was separated carefully, using tools made of insulating material
in order not to disturb the induced charge on it, and the negative induced charge
on each hemisphere was measured.
Faraday found that the total charge on the outer sphere was equal in magnitude to
the original charge placed on the inner sphere and that this was true regardless of the
dielectric material separating the two spheres. He concluded that there was some sort
of “displacement” from the inner sphere to the outer which was independent of the
medium, and we now refer to this flux as displacement, displacement flux, or simply
electric flux.
Faraday’s experiments also showed, of course, that a larger positive charge on the
inner sphere induced a correspondingly larger negative charge on the outer sphere,
leading to a direct proportionality between the electric flux and the charge on the inner
sphere. The constant of proportionality is dependent on the system of units involved,
and we are fortunate in our use of SI units, because the constant is unity. If electric
flux is denoted by _ (psi) and the total charge on the inner sphere by Q, then for
Faraday’s experiment
and the electric flux _ is measured in coulombs.
Electric flux density, measured in coulombs per square meter (sometimes described
as “lines per square meter,” for each line is due to one coulomb), is given
the letter D, which was originally chosen because of the alternate names of displacement
flux density or displacement density.
The electric flux density D is a vector field and is a member of the “flux density”
class of vector fields, as opposed to the “force fields” class, which includes the electric field
intensity E. The direction of D at a point is the direction of the flux lines at that
point, and the magnitude is given by the number of flux lines crossing a surface normal
to the lines divided by the surface area.
CURRENT DENSITY:
Electric charges in motion constitute a current. The unit of current is the ampere (A), defined as
a rate of movement of charge passing a given reference point (or crossing a given reference
plane) of one coulomb per second. Current is symbolized by I , and therefore
Current is thus defined as the motion of positive charges, even though conduction in metals takes
place through the motion of electrons.
.
In field theory, we are usually interested in events occurring at a point rather than within a large
region, and we find the concept of current density, measured in amperes per square meter
(A/m2
), more useful. Current density is a vector represented by
DIVERGENCE:
The divergence of A is defined as
and is usually abbreviated div A. The physical interpretation of the divergence of a
vector is obtained by describing carefully the operations implied by the right-hand
side of above equation.
“The divergence of the vector flux density A is the outflow of flux from a small closed surface
per unit volume as the volume shrinks to zero.Divergence at a point (x,y,z) is the measure of
the vector flow out of a surface surrounding that point”.
That is, imagine a vector field represents water flow. Then if the divergence is a positive
number, this means water is flowing out of the point (like a water spout - this location is
considered a source). If the divergence is a negative number, then water is flowing into the point
(like a water drain - this location is known as a sink).
First, imagine we have a vector field as shown in Figure, and we want to know what the
divergence is at the point P:
MAXWELL’S FIRST EQUATION
Gauss' Law is the first of Maxwell's Equations which dictates how the Electric Field behaves
around electric charges. Gauss' Law can be written in terms of the Electric Flux Density and
the Electric Charge Density as:
[Equation 1]
In Equation [1], the symbol is the divergence operator.
Equation [1] is known as Gauss' Law in point form. That is, Equation [1] is true at any point in
space. That is, if there exists electric charge somewhere, then the divergence of D at that point is
nonzero, otherwise it is equal to zero.
We have already explained that divergence states that what is left behind when a vector passes a
certain point. This equation states that the source of electric flux is electric charge.Whenever
electric charge is there ,the divergence has non zero value which is equal to the volume charge
density.
According to this total electric flux through any closed surface is times the total charge
enclosed by the closed surfaces, representing Gauss's law of electrostatics, As this does not
depend on time, it is a steady state equation. Here for positive , divergence of electric field is
positive and for negative , divergence is negative. It indicates that is scalar quantity
To get some more intuition on Gauss' Law, let's look at Gauss' Law in integral form. To do this,
we assume some arbitrary volume (we'll call it V) which has a boundary (which is written S).
Then integrating Equation [1] over the volume V gives Gauss' Law in integral form:
MAXEWELL’S SECOND EQUATION
Second equation is the statement of Gauss law in magnetic field. It states that:
As the divergence indicates the source, so this equation states that isolated magnetic poles
or magnetic monopoles cannot exist as they appear only in pairs and there is no source or sink
for magnetic lines of forces. It is also independent of time i.e. steady state equation.This is
illustrated in following figure:
Magnets attract other magnets similar to how electric charges repel or attract like electric
charges. However, there is something special about these magnets - they always have a positive
and negative end. This means every magnetic object is a magnetic dipole, with a north and south
pole. No matter how many times you break the magnetic in half, it will just form more magnetic
dipoles. Gauss' Law for Magnetism states that magnetic monopoles do not exist - or at least we
haven't found them yet.
Because we know that the divergence of the Magnetic Flux Density is always zero, we now
know a little bit about how these fields behave. We will present a couple of examples of legal
and illegal Magnetic Fields, which are a consequence of Gauss' Law for Magnetism:
MAXWELL’S THIRD EQUATION:
Third equation is simply the statement of Faraday law. It states that:
According to this equation, a magnetic field that is changing in time will give rise to a circulating
E-field. This means we have two ways of generating E-fields - from Electric Charges (or flowing
electric charge, current) or from a magnetic field that is changing.
Faraday was a scientist experimenting with circuits and magnetic coils way back in the 1830s.
His experiment setup, which led to Farday's Law
The experiment itself is somewhat simple. When the battery is disconnected, we have no electric
current flowing through the wire. Hence there is no magnetic flux induced within the Iron
(Magnetic Core). The Iron is like a highway for Magnetic Fields - they flow very easily through
magnetic material. So the purpose of the core is to create a path for the Magnetic Flux to flow.
When the switch is closed, the electric current will flow within the wire attached to the battery.
When this current flows, it has an associated magnetic field (or magnetic flux) with it. When the
wire wraps around the left side of the magnetic cor, a magnetic field (magnetic flux) is induced
within the core. This flux travels around the core. So the Magnetic Flux produced by the wired
coil on the left exists within the wired coil on the right, which is connected to the ammeter.
Now, a funny thing happens, which Faraday observed. When he closed the switch, then current
would begin flowing and the ammeter would spike one way. But this was very brief, and the
current on the right coil would go to zero. When the switch was opened, the measured current would spike to the other side , and then the measured current on the right side would again be
zero.
Faraday figured out what was happening. When the switch was initially changed from open to
closed, the magnetic flux within the magnetic core increased from zero to some maximum
number (which was a constant value, versus time). When the flux was increasing, there existed
an induced current on the opposite side.
Similarly, when the switch was opened, the magnetic flux in the core would decrease from its
constant value back to zero. Hence, a decreasing flux within the core induced an opposite current
on the right side.
Faraday figured out that a changing Magnetic Flux within a circuit (or closed loop of wire)
produced an induced EMF, or voltage within the circuit. He wrote this as
MAXWELL’S FOURTH EQUATION:
This equation is the statement of Ampere’s circuital law with the extension of a concept called
DISPLACEMENT CURRENT”. This was devised by Maxwell to account for the varying fields.
This states that:
[Equation 1]
Ampere was a scientist experimenting with forces on wires carrying electric current. He was
doing these experiments back in the 1820s, about the same time that Farday was working
on Faraday's Law. Ampere and Farday didn't know that there work would be unified by Maxwell
himself, about 4 decades later.
Ampere's Law relates an electric current flowing and a magnetic field wrapping around it:
[Equation 2]
Equation [2] can be explained: Suppose you have a conductor (wire) carrying a current, I. Then
this current produces a Magnetic Field which circles the wire.
The left side of Equation [2] means: If you take any imaginary path that encircles the wire, and
you add up the Magnetic Field at each point along that path, then it will numerically equal the
amount of current that is encircled by this path (which is why we write for encircled or
enclosed current).
Let's do an example for fun. Suppose we have a long wire carrying a constant electric
current, I[Amps]. What is the magnetic field around the wire, for any distance r [meters] from
the wire?
Consider the following figure. We have a long wire carrying a current of I Amps. We want to
know what the Magnetic Field is at a distance r from the wire. So we draw an imaginary path
around the wire, which is the dotted blue line .