Electromagnetic Theory
Vector Calculus
COORDINATE SYSTEMS • RECTANGULAR or Cartesian
• CYLINDRICAL • SPHERICAL
Examples: Sheets - RECTANGULAR Wires/Cables - CYLINDRICAL Spheres - SPHERICAL
Choice is based on symmetry of problem
Cartesian Coordinates Or Rectangular Coordinates
P (x, y, z)
z
x
P(x,y,z)
y
y
z x
A vector A in Cartesian coordinates can be written as
( Ax , Ay , Az )
or
Ax ax Ay a y Az az
where ax,ay and az are unit vectors along x, y and z-directions.
Cylindrical Coordinates P (ρ, Φ, z)
0 0 2 z
A vector A in Cylindrical coordinates can be written as
( A , A , Az )
or
A a A a Az az
where aρ,aΦ and az are unit vectors along ρ, Φ and z-directions. x= ρ cos Φ, y=ρ sin Φ,
z=z
y x y , tan ,z z x 2
2
1
Spherical Coordinates
P (r, θ, Φ)
0r 0 0 2
A vector A in spherical coordinates can be written as
( Ar , A , A )
or
Ar ar A a A a
where ar, aθ, and aΦ are unit vectors along r, θ, and Φ-directions. x=r sin θ cos Φ, y=r sin θ sin Φ,
r x 2 y 2 z 2 , tan 1
Z=r cos θ
x2 y 2 1 y , tan z x
Differential Length, Area and Volume
Cartesian Coordinate system
Cartesian Coordinate System 1- Differential displacement is given by
dl dxax dyay dzaz 2- Differential normal surface area is given by dS dydzax dS dxdza y dS dxdyaz
3- Differential volume is given by-
dv dxdydz
Cylindrical Coordinate System
Cylindrical Coordinate System 1- Differential displacement is given by
dl d a d a dzaz
2- Differential normal surface area is given by dS d dza
dS d dza dS d d az 3- Differential volume is given by-
dv d d dz
Spherical Coordinate System
Spherical Coordinate System 1- Differential displacement is given by dl drar rd a r sin d a 2- Differential normal surface area is given by
dS r sin d d ar 2
dS r sin drd a dS rdrd a 3- Differential volume is given by-
dv r sin drd d 2
Scalar and Vector Fields • A scalar field is a function that gives us a single value (of magnitude only) of some variable for every point in
space. voltage, current, energy, temperature
• A vector is a quantity which has both a magnitude and a direction in space. velocity, momentum, acceleration and force
Del Operator “a vector differentiation operator” Cartesian Coordinates
ax a y az x y z
Cylindrical Coordinates
1 a a az z Spherical Coordinates
1 1 ar a a r r r sin
Gradient, Divergence and Curl The Del Operator ax a y az x y z • Gradient of a scalar function is a vector quantity. • Divergence of a vector is a scalar quantity. • Curl of a vector is a vector quantity.
• The Laplacian of a scalar A is a scalar quantity.
f . A A
A 2
Gradient of a Scalar The gradient of a scalar field V is a vector that represents both the magnitude and the direction of the space rate of
change of V.
V V V V ax ay az x y z
Divergence of a Vector The divergence of a vector A at a given point P is the volume density of the outward flux of vector field of A from an infinitesimal volume around a given point P.
divA . A A A A . A x y z
Physical Significance: The divergence measures how much a vector field ``spreads out'' or diverges from a given point. If the vector field lines are coming outwards from the given point----- Divergence is positive. If the vector field lines are coming in towards the point----Divergence is negative.
Curl of a Vector The curl of A is an axial vector whose magnitude is the maximum circulation of A per unit area and whose direction is the normal direction of the area.
ax A x Ax
ay y Ay
az z Az
Physical Significance of Curl The curl of a vector field measures the tendency for the vector field to rotate around the point. If the value of curl is zero then the field is said to be a non rotational field (in such condition, divergence will have non-zero value)
Divergence Theorem or Gauss’ Theorem The divergence theorem states that the total outward flux of a vector field A through the closed surface S is the same
as the volume integral of the divergence of A.
A.dS .Adv S
V
Stokes’ Theorem Stokes’s theorem states that the line integral of a vector field A around a closed path L is equal to the surface integral of the curl of A over the open surface S bounded by L, provided A and
A are continuous on S
A.dl ( A).dS L
S
Electric Field Intensity Electric Field Intensity is the force per unit charge when placed in the electric field
F E Q
In general, if we have more than two point charges
Electric Field due to Continuous Charge Distribution If there is a continuous charge distribution say along a line, on a surface, or in a volume
The charge element dQ and the total charge Q due to these charge distributions can be obtained by
The electric field intensity due to each charge distribution ρL, ρS and ρV may be given by the summation of the field contributed by the numerous point charges making up the charge distribution.
The value of R and aR vary as we evaluate above three integrals.
Electric Flux Density The electric field intensity E depends on the medium in which the charges are placed.
D oE The vector field D is called the electric displacement vector or electric flux density and is measured in coulombs per square meter.
The electric flux ψ in terms of D can be defined as
s
Gauss Law It states that the total electric flux ψ through any closed surface is equal to the total charge enclosed by that surface.
Qenc
(i)
Using Divergence Theorem (ii) Comparing the two volume integrals in (i) and (ii)
It states that the volume charge density is the same as the divergence of the electric flux density. (This is also the first Maxwell’s equation.)
Relationship between E and V The potential difference across a closed path is zero, i. e.
VAB VBA B
A
A
B
VAB E.d l and VBA E.d l
VAB VBA E.d l 0
E.d l 0 It means that the line integral of
(i)
E along a closed path must be zero.
Physically it means that no net work is done in moving a charge along a closed path in an electrostatic field. Applying Stokes’s theorem to equation (i)
E.d l ( E).d S 0 E 0
(ii)
Equation (i) and (ii) are known as Maxwell’s equation for static electric fields. Equation (i) is in integral form while equation (ii) is in differential form, both depicting conservative nature of an electrostatic field.
Electric Potential due to an Electric Dipole An electric dipole is formed when two point charges of equal magnitude but of opposite sign are separated by a small distance. The potential at P (r, θ, Φ) is
If r >> d, r2 - r1 = d cosθ and r1r2 = r2 then
But
d cos d .ar
where d d az
If we define p Qd as the dipole moment, then
The dipole moment p is directed from –Q to +Q. if the dipole center is not at the origin but at
r ' then
Polarization in Dielectrics • A dielectric substance is an electrical insulator that can be polarized by an applied electric field. • When a dielectric is placed in an electric field, electric charges do not flow through the material (as they do in a conductor), but they only slightly shift from their average equilibrium positions causing dielectric polarization. • Because of dielectric polarization, positive charges are displaced toward the field and negative charges shift in the opposite direction. • This creates an internal electric field which reduces the overall field within the dielectric itself.
For the measurement of intensity of polarization, we define polarization P as dipole moment per unit volume (Unit: coulomb per square meter)
The major effect of the electric field on the dielectric is the creation of dipole moments that align themselves in the direction of electric field.
This type of dielectrics are said to be non-polar. eg: H2, N2, O2
Other types of molecules that have in-built permanent dipole moments are called polar. eg: H2O, HCl When electric field is applied to a polar material then its permanent dipole experiences a torque that tends to align its dipole moment in the direction of the electric field.
Field due to a Polarized Dielectric Consider a dielectric material consisting of dipoles with Dipole moment P per unit volume. The potential dV at an external point O due to Pdv
'
(i)
where R is the distance between volume element dv’ and the point O, i.e. R2 = (x-x’)2+(y-y’)2+(z-z’)2 But (Vector Identity) Here primed quantities are having primed del operator
Now, since or
=
-
Put this in (i) and integrate over the entire volume v’ of the dielectric
Applying Divergence Theorem to the first term (ii) where an’ is the outward unit normal to the surface dS’ of the dielectric The two terms in (ii) denote the potentials due to surface and volume charge distributions with densities given as-
Where ρps = Bound surface charge density and ρpv = Volume charge density. Bound charges are those which are not free to move in the dielectric material. Free charges are those that are capable of moving over macroscopic distance. (The charge due to polarization is known as bound charge, while charge on an object produced by electrons gained or lost from outside the object is called free charge. )
Whenever, polarization occurs, an equivalent volume charge density, ρpv is formed throughout the dielectric while an equivalent surface charge density, ρps is formed over the surface of dielectric.
Now, let us take the case when dielectric contains free charge If ρv is the free volume charge density then the total volume charge density ρt
Hence
Where
Dielectric Constant and Strength No dielectric is ideal. When the electric field in a dielectric is sufficiently high then it begins to pull electrons completely out of the molecules, and the dielectric starts conducting. If ρv is the free volume charge density then the total volume charge density ρt Here
Hence
Where
= Charge density arising due to polarization (bound charges)
The effect of the dielectric on electric field E , is to increase D inside it by an amount P . Further, the polarization would vary directly as the applied electric field. So one can write-
Where e is known as the electric susceptibility of the material It is a measure of how susceptible a given dielectric is to electric fields.
We know that and Thus or where
o r
and
where є is the permittivity of the dielectric, єo is the permittivity of the free space and єr is the dielectric constant or relative permittivity.
When a dielectric becomes conducting then it is called dielectric breakdown. It depends on the type of material, humidity, temperature and the amount of time for which the field is applied. The minimum value of the electric field at which the dielectric breakdown occurs is called the dielectric strength of the dielectric material. or The dielectric strength is the maximum value of the electric field that a dielectric can tolerate or withstand without breakdown.
Continuity Equation and Relaxation Time According to principle of charge conservation, the time rate of decrease of charge within a given volume must be equal to the net outward current flow through the closed surface of the volume. The current Iout coming out of the closed surface (i) where Qin is the total charge enclosed by the closed surface. Using divergence theorem
But
Equation (i) now becomes
or
(ii)
This is called the continuity of current equation. Effect of introducing charge at some interior point of a conductor/dielectric According to Ohm’s law
According to Gauss’s law
Equation (ii) now becomes
or This is homogeneous linear ordinary differential equation. By separating variables we get
Integrating both sides
(iii) where
Equation (iii) shows that as a result of introducing charge at some interior point of the material there is a decay of the volume charge density ρv. The time constant Tr is known as the relaxation time . Relaxation time is the time in which a charge placed in the interior of a material to drop to e-1 = 36.8 % of its initial value.
For Copper Tr = 1.53 x 10-19 sec (Tr -short for good conductors) For fused Quartz Tr = 51.2 days (Tr -large for good dielectrics)
Electric Field: Boundary Conditions If the electric field exists in a region consisting of two different media, the conditions that the field must satisfy at the interface separating the media are called boundary conditions These conditions are helpful in determining the field on one side of the boundary when the field on other side is known.
We will consider the boundary conditions at an interface separating 1. Dielectric (єr1) and Dielectric (єr2) 2. Conductor and Dielectric 3. Conductor and free space For determining boundary conditions we will use Maxwell’s equations and
Boundary Conditions
• Also we need to decompose E to the interface of the interestE Et En
Where En and Et are the normal and tangential components of E to the interface of the medium
Dielectric- Dielectric boundary conditions • Let us consider the E field exists in a region that consists of two different dielectrics characterized by1 0 r1and 2 0 r 2 as shown below-
In media 1 and 2, Electric field can be decomposed as
Using Maxwell Equation
along the closed path abcda-
As ∆h→0, above equation becomes
----------------- (i) Thus the tangential components are the same on the two sides of the boundary, i.e Et is continuous across the boundary.
Since
D E Dn Dt
Thus, Dt undergoes some change, hence it is said to be discontinuous across the interface.
We can also apply first Maxwell equation, to the Cylindrical Gaussian Surface of the adjacent figure.
The contribution due to curved sides vanishes. Allowing ∆h→0 gives
or ----------------- (ii) here is the free charge density at boundary
• Generally = 0 (until and unless, we are not putting free charge at the interface deliberately), the above equation become----------------- (iii)
• Thus the normal component of D is continuous across the interface. Further since D E, we can write----------------- (iv)
• Showing that the normal component of the E is discontinuous at the boundary. • Equations (i), (ii), (iii) and (iv) are known as BOUNDARY CONDITIONS. (They must be satisfied at the dielectric- dielectric boundary)
