EMT UNIT-1 Q.1 What is the gauss’s law

EMT
UNIT-1
Q.1 What is the gauss’s law
Gauss's law may be expressed as:
where ΦE is the electric flux through a closed
surface S enclosing any volume V, Q is the
total charge enclosed within S, and ε0 is theelectric constant.
The electric flux ΦE is defined as a surface integral of
the electric field:
where E is the electric field, dA is a vector representing
an infinitesimal element of area,[note 1] and • represents
the dot product of two vectors.
Since the flux is defined as an integral of the electric field,
this expression of Gauss's law is called the integral form.
[edit]Applying the integral form
Introduction
The electric field of a given charge distribution can in
principle be calculated using Coulomb's law. The examples
discussed in Chapter 23 showed however, that the actual
calculations can become quit complicated. Gauss' Law
An alternative method to calculate the electric field of a given
charge distribution relies on a theorem called Gauss' law. Gauss'
law states that
" If the volume within an arbitrary closed mathematical surface
holds a net electric charge Q, then the electric flux [Phi] though
its surface is Q/[epsilon]0 "
Gauss' law can be written in the following form:
Figure 24.1. Electric flux through surface area A.
The electric flux [Phi] through a surface is defined as the
product of the area A and the magnitude of the normal
component of the electric field E:
where [theta] is the angle between the electric field and the
normal of the surface (see Figure 24.1). To apply Gauss' law one
has to obtain the flux through a closed surface. This flux can be
obtained by integrating eq.(24.2) over all the area of the surface.
The convention used to define the flux as positive or negative is
that the angle [theta] is measured with respect to the
perpendicular erected on the outside of the closed surface: field
lines leaving the volume make a positive contribution, and field
lines entering the volume make a negative contribution.
Example
Charge is uniformly distributed over the volume of a
large slab of plastic of thickness d. The charge density is [rho]
C/m3. The mid-plane of the slab is the y-z plane (see Figure
24.3). What is the electric filed at a distance x from the midplane ?
Figure Problem
As a result of the symmetry of the slab, the direction of the
electric field will be along the x-axis (at every point). To
calculate the electric field at any given point, we need to
consider two separate case: - d/2 < x < d/2 and x > d/2 or x < d/2. Consider surface 1 shown in Figure 24.3. The flux through
this surface is equal to the flux through the planes at x = x1 and x
= - x1. Symmetry arguments show that
The flux [Phi]1 through surface 1 is therefore given by
The amount of charge enclosed by surface 1 is given by
Applying Gauss' law to eq.(24.7) and eq.(24.8) we obtain
or
Note: this formula is only correct for - d/2 < x1 < d/2.
The flux [Phi]2 through surface 2 is given by
The charge enclosed by surface 2 is given by
This equation shows that the enclosed charge does not depend
on x2. Applying Gauss's law one obtains
or
Conductors in Electric Fields
A large number of electrons in a conductor are free to move.
The so called free electrons are the cause of the different
behavior of conductors and insulators in an external electric
filed. The free electrons in a conductor will move under the
influence of the external electric field (in a direction opposite to
the direction of the electric field). The movement of the free
electrons will produce an excess of electrons (negative charge)
on one side of the conductor, leaving a deficit of electrons
(positive charge) on the other side. This charge distribution will
also produce an electric field and the actual electric field inside
the conductor can be found by superposition of the external
electric field and the induced electric field, produced by the
induced charge distribution. When static equilibrium is reached,
the net electric field inside the conductor is exactly zero. This
implies that the charge density inside the conductor is zero. If
the electric field inside the conductor would not be exactly zero
the free electrons would continue to move and the charge
distribution would not be in static equilibrium. The electric field
on the surface of the conductor is perpendicular to its surface. If
this would not be the case, the free electrons would move along
the surface, and the charge distribution would not be in
equilibrium. The redistribution of the free electrons in the
conductor under the influence of an external electric field, and
the cancellation of the external electric field inside the conductor
is being used to shield sensitive instruments from external
electric fields.
The strength of the electric field on the surface of a conductor
can be found by applying Gauss' law (see Figure 24.4). The
electric flux through the surface shown in Figure 24.4 is given
by
where A is the area of the top of the surface shown in Figure
24.4. The flux through the bottom of the surface shown in
Figure 24.4 is zero since the electric field inside a conductor is
equal to zero. Note that eq.(24.15) is only valid close to the
conductor where the electric field is perpendicular to the surface.
The charge enclosed by the surface shown in Figure 24.4 is
equal to
Figure 24.4. Electric field of conductor.
where [sigma] is the surface charge density of the conductor.
Eq.(24.16) is correct if the charge density [sigma] does not vary
significantly over the area A (this condition can always be met
by reducing the size of the surface being considered). Applying
Gauss' law we obtain
Thus, the electric filed at the surface of the conductor is given
by
Q.2 what is the divergence theorem?
The divergence theorem, more commonly known especially in
older literature as Gauss's theorem and also known as the
Gauss-Ostrogradsky theorem, is a theorem in vector calculus
that can be stated as follows. Let be a region in space with
boundary . Then the volume integral of the divergence
of
over and the surface integral of over the boundary of are
related by
(1)
The divergence theorem is a mathematical statement of the
physical fact that, in the absence of the creation or destruction of
matter, the density within a region of space can change only by
having it flow into or away from the region through its
boundary.
A special case of the divergence theorem follows by specializing
to the plane. Letting be a region in the plane with boundary ,
equation (1) then collapses to
(2)
If the vector field satisfies certain constraints, simplified forms
can be used. For example, if
where is a constant
vector , then
(3)
But
(4)
so
(5)
(6)
and
(7)
But
, and
must vary with so that
equal zero. Therefore,
cannot always
(8)
Similarly, if
then
, where is a constant vector
,
(9)
Q.3 what is the stoke Theorem ?
Introduction
The fundamental theorem of calculus states that the integral of a
function f over the interval [a, b] can be calculated by finding
anantiderivative F of f:
Stokes' theorem is a vast generalization of this theorem in the
following sense.
By the choice of F,
. In the parlance of differential
forms, this is saying that f(x) dx is the exterior derivative of
the 0-form, i.e. function, F: in other words, that dF = f dx.
The general Stokes theorem applies to higher differential
forms instead of F.
A closed interval [a, b] is a simple example of a onedimensional manifold with boundary. Its boundary is the
set consisting of the two points a and b. Integrating f over
the interval may be generalized to integrating forms on a
higher-dimensional manifold. Two technical conditions are
needed: the manifold has to be orientable, and the form has
to be compactly supported in order to give a well-defined
integral.
The two points a and b form the boundary of the open
interval. More generally, Stokes' theorem applies to
oriented manifolds M with boundary. The boundary
∂M of M is itself a manifold and inherits a natural
orientation from that of the manifold. For example, the
natural orientation of the interval gives an orientation of
the two boundary points. Intuitively, a inherits the opposite
orientation as b, as they are at opposite ends of the interval.
So, "integrating" F over two boundary points a, b is taking
the difference F(b) − F(a).
In even simpler terms, one can consider that points can be
thought of as the boundaries of curves, that is as 0dimensional boundaries of 1-dimensional manifolds. So, just
as one can find the value of an integral (f dx = dF) over a 1dimensional manifolds ([a,b]) by considering the antiderivative (F) at the 0-dimensional boundaries ([a,b]), one can
generalize the fundamental theorem of calculus, with a few
additional caveats, to deal with the value of integrals (dω)
over n-dimensional manifolds (Ω) by considering the anti-
derivative (ω) at the (n-1)-dimensional boundaries (dΩ) of the
manifold.
So the fundamental theorem reads:
Q.4 what is the curl law ?
A special case of Stokes' theorem in which is a vector
field and is an oriented, compact embedded 2-manifold with
boundary in , and a generalization of Green's theorem from the
plane into three-dimensional space. The curl theorem states
(1)
where the left side is a surface integral and the right side is a line
integral.
There are also alternate forms of the theorem. If
(2)
then
(3)
and if
(4)
then
(5)
A special case of Stokes' theorem in which is a vector
field and is an oriented, compact embedded 2-manifold with
boundary in , and a generalization of Green's theorem from the
plane into three-dimensional space. The curl theorem states
(1)
where the left side is a surface integral and the right side is a line
integral.
There are also alternate forms of the theorem. If
(2)
then
(3)
and if
(4)
then
(5)
A special case of Stokes' theorem in which is a vector
field and is an oriented, compact embedded 2-manifold with
boundary in , and a generalization of Green's theorem from the
plane into three-dimensional space. The curl theorem states
(1)
where the left side is a surface integral and the right side is a line
integral.
There are also alternate forms of the theorem. If
(2)
then
(3)
and if
(4)
then
(5)
Q.5 prove the relation between current density and volume
charge denaity
Continuous charges
Continuous charge distribution. The volume charge density ρ is
the amount of charge per unit volume (cube), surface charge
density σ is amount per unit surface area (circle) with outward
unit normal n, d is the dipole moment between two point
charges, the volume density of these is the polarization density
P. Position vector r is a point to calculate the electric field; r′ is a
point in the charged object.
Following are the definitions for continuous charge
distributions.[2][3]
The linear charge density is the ratio of an infinitesimal electric
charge dQ (SI unit: C) to an infinitesimal line element,
similarly the surface charge density uses a surface area element
dS
and the volume charge density uses a volume element dV
Integrating the definitions gives the total charge Q of a region
according to line integral of the linear charge density λq(r) over a
line or 1d curve C,
similarly a surface integral of the surface charge density σq(r)
over a surface S,
and a volume integral of the volume charge density ρq(r) over a
volume V,
where the subscript q is to clarify that the density is for electric
charge, not other densities like mass density, number density,
probability density, and prevent conflict with the many other
uses of λ, σ, ρ in electromagnetism for wavelength, electrical
resistivity and conductivity.
Within the context of electromagnetism, the subscripts are
usually dropped for simplicity: λ, σ, ρ. Other notations may
include: ρℓ, ρs, ρv, ρL, ρS, ρV etc.
Average charge densities
The total charge divided by the length, surface area, or volume
will be the average charge densities:
Q.6 DEFINE AND PROVE BIOT SAVSRT LAW.
In physics, particularly electromagnetism, the Biot–Savart law
(pron.: /ˈbiˈoʊ səˈvɑr/ or /ˈbjoʊ səˈvɑr/)[1] is an equation that
describes the magnetic field generated by an electric current. It
relates the magnetic field to the magnitude, direction, length,
and proximity of the electric current. The law is valid in the
magnetostatic approximation, and is consistent with both
Ampère's circuital law and Gauss's law for magnetism.[2]
Electric currents (along closed curve)
The Biot–Savart law is used to compute the resultant magnetic
field B at position r generated by a steady current I (for example
due to a wire): a continual flow of charges which is constant in
time and the charge neither accumulates nor depletes at any
point. The law is a physical example of a line integral: evaluated
over the path C the electric currents flow. The equation in SI
units is[3]
where r is the full displacement vector from the wire element to
the point at which the field is being computed and r is the unit
vector of r. Using this the equation can be equivalently written
where dl is a vector whose magnitude is the length of the
differential element of the wire, in the direction of conventional
current, and μ0 is the magnetic constant. The symbols in
boldface denote vector quantities.
The integral is usually around a closed curve, since electric
currents can only flow around closed paths. An infinitely long
wire (as used in the definition of the SI unit of electric current the Ampere) is a counter-example.
To apply the equation, the point in space where the magnetic
field is to be calculated is chosen. Holding that point fixed, the
line integral over the path of the electric currents is calculated to
find the total magnetic field at that point. The application of this
law implicitly relies on the superposition principle for magnetic
fields, i.e. the fact that the magnetic field is a vector sum of the
field created by each infinitesimal section of the wire
individually.[4]
Electric currents (throughout conductor volume)
The formulations given above work well when the current can
be approximated as running through an infinitely-narrow wire. If
the current has some thickness, the proper formulation of the
Biot–Savart law (again in SI units) is:
or equivalently
where dV is the differential element of volume and J is the
current density vector in that volume.
In this case the integral is over the volume of the conductor.
The Biot–Savart law is fundamental to magnetostatics, playing a
similar role to Coulomb's law in electrostatics. When
magnetostatics does not apply, the Biot–Savart law should be
replaced by Jefimenko's equations.
Constant uniform current
In the special case of a steady constant current I, the magnetic
field B is
i.e. the current can be taken out the integral.
Point charge at constant velocity
In the case of a point charged particle q moving at a constant
velocity v, then Maxwell's equations give the following
expression for the electric field and magnetic field:[5]
where r is the vector pointing from the current (non-retarded)
position of the particle to the point at which the field is being
measured, and θ is the angle between v and r.
When v2 ≪ c2, the electric field and magnetic field can be
approximated as[5]
These equations are called the "Biot–Savart law for a point
charge"[6] due to its closely analogous form to the "standard"
Biot–Savart law given previously. These equations were first
derived by Oliver Heaviside in 1888.
Magnetic responses applications
The Biot–Savart law can be used in the calculation of magnetic
responses even at the atomic or molecular level, e.g. chemical
shieldings or magnetic susceptibilities, provided that the current
density can be obtained from a quantum mechanical calculation
or theory.
Aerodynamics applications
The figure shows the velocity ( ) induced at a point P by an
element of vortex filament ( ) of strength .
The Biot–Savart law is also used in aerodynamic theory to
calculate the velocity induced by vortex lines.
In the aerodynamic application, the roles of vorticity and current
are reversed as when compared to the magnetic application.
In Maxwell's 1861 paper 'On Physical Lines of Force',[7]
magnetic field strength H was directly equated with pure
vorticity (spin), whereas B was a weighted vorticity that was
weighted for the density of the vortex sea. Maxwell considered
magnetic permeability μ to be a measure of the density of the
vortex sea. Hence the relationship,
1. Magnetic induction current
was essentially a rotational analogy to the linear electric
current relationship,
2. Electric convection current
where ρ is electric charge density. B was seen as a kind of
magnetic current of vortices aligned in their axial planes,
with H being the circumferential velocity of the vortices.
The electric current equation can be viewed as a convective
current of electric charge that involves linear motion. By
analogy, the magnetic equation is an inductive current involving
spin. There is no linear motion in the inductive current along the
direction of the B vector. The magnetic inductive current
represents lines of force. In particular, it represents lines of
inverse square law force.
In aerodynamics the induced air currents are forming solenoidal
rings around a vortex axis that is playing the role that electric
current plays in magnetism. This puts the air currents of
aerodynamics into the equivalent role of the magnetic induction
vector B in electromagnetism.
In electromagnetism the B lines form solenoidal rings around
the source electric current, whereas in aerodynamics, the air
currents form solenoidal rings around the source vortex axis.
Hence in electromagnetism, the vortex plays the role of 'effect'
whereas in aerodynamics, the vortex plays the role of 'cause'.
Yet when we look at the B lines in isolation, we see exactly the
aerodynamic scenario in so much as that B is the vortex axis and
H is the circumferential velocity as in Maxwell's 1861 paper.
For a vortex line of infinite length, the induced velocity at a
point is given by
where Γ is the strength of the vortex and r is the perpendicular
distance between the point and the vortex line.
This is a limiting case of the formula for vortex segments of
finite length
where A and B are the (signed) angles between the line and the
two ends of the segment.
Cylindrical Capacitor
The capacitance for cylindrical
orspherical conductors can be obtained by
evaluating the voltage difference between the
conductors for a given charge on each. By
applying Gauss' law to an infinite cylinder in a
vacuum, the electric field outside a charged
cylinder is found to be
Index
Capacitor
Concepts
The voltage between the cylinders can be found
by integrating the electric field along a radial
line:
From the definition of capacitance and including
the case where the volume is filled by a
dielectric of dielectric constant k, the
capacitance per unit length is defined as
Calculation