UPH004/013
Electromagnetic waves
Department of Physics and Materials Science (DPMS)
Thapar Institute of Engineering and Technology
1. FIELD ???
Field is the region of influence by source of the effect.
In other words the field is a multi dimensional function. This
function may be scalar to give scalar field or it may be a
vector to give vector field.
The examples of scalar field may be the distribution of
temperature in a room, the potential energy around a charge
distribution.
The examples of vector field may be the electric or magnetic
field around charge or current distribution, respectively.
2. GRADIENT OF A SCALAR Field
For the present case let the temperature distribution in a room.
The temperature T(x,y,z) forms a scalar field.
Suppose T1(x,y,z) is the temperature at point P1(x,y,z), and
T2(x+dx,y+dy,z+dz) is the temperature at another point
P2(x+dx,y+dy,z+dz) as shown.
2. GRADIENT OF A SCALAR field
The differential distances dx, dy , dz are the components of the
differential distance vector dL:
dL = dxiˆ + dyˆj + dzk̂
However, from differential calculus, the differential
temperature:
∂T ∂T ∂T
dT = T2 − T1 = dx + dy + dz
∂x ∂y ∂z
dx = dL • iˆ
dy = dL • ĵ
dz = dL • k̂
2. GRADIENT OF A SCALAR field
So, previous equation can be rewritten as:
∂T ∂T ˆ ∂T ˆ
dT = î • dL + j • dL + k • dL
∂x ∂y ∂z
∂T ∂T ˆ ∂T
= î + j+ kˆ • dL
∂x ∂y ∂z
The vector inside square brackets defines the change of
temperature dT corresponding to a vector change in position dL.
This vector is called Gradient of Scalar T.
For Cartesian coordinate:
∂T ∂T ∂T
∇T = î + ĵ + k̂
∂x ∂y ∂z
2. GRADIENT OF A SCALAR field
As one of the example you may think of a hill. The height H of
the hill is a function of two coordinates say x and y. H(x,y).
The gradient of height of the hill is related with the slop of the
hill at the point of question.
# Del “operator” #
It can be written also as:
∂ ∂ ∂
∇T = î + ĵ + k̂ T
∂x ∂y ∂z
By looking at this one may ponder:
∂ ∂ ∂
∇ ≡ î + ĵ + k̂
∂x ∂y ∂z
The operator ∇ is called as “del” operator.
It is not a vector however, it behaves like a vector in terms of dot
product and cross product.
3. DIVERGENCE OF A VECTOR
Illustration of the divergence of a vector field at point P:
Positive Negative Zero
Divergence Divergence Divergence
∂Ax ∂Ay ∂Az
∇•A = + +
∂x ∂y ∂z
4. CURL OF A VECTOR
For Cartesian coordinate: iˆ ˆj kˆ
∂ ∂ ∂
∇×A =
∂x ∂y ∂z
Ax Ay Az
∂Az ∂Ay ∂Az ∂Ax ∂Ay ∂Ax
∇×A = − î − ∂x − ĵ + − k̂
∂y ∂z ∂z ∂x ∂y
4. CURL OF A VECTOR
The curl of the vector field is concerned with rotation of the
vector field. Rotation can be used to measure the uniformity of
the field, the more non uniform the field, the larger value of curl.
5. LAPLACIAN
The Laplacian of a scalar field, T written as: ∇ 2T
Where, Laplacian T is:
∇ 2T = ∇ • ∇T
∂ ∂ ˆ ∂ ˆ ∂T ∂T ˆ ∂T
= iˆ + j + k • î + j+ kˆ
∂x ∂y ∂z ∂x ∂y ∂z
∂ 2
T ∂ 2
T ∂ 2
T
∇T= 2 + 2 + 2
2
∂x ∂y ∂z
The Laplacian of a scalar field, V written as:
(
∇ V = ∇ 2 Vx iˆ + V y ˆj + Vz kˆ
2
)
( ) ( ) (
= ∇ 2Vx iˆ + ∇ 2V y ˆj + ∇ 2Vz kˆ)
Higher order derivatives
∇ × (∇T ) = 0
∇ • (∇ × A) = 0
∇( A • B ) = A × (∇ × B ) + B × (∇ × A) + ( A • ∇) B + ( B • ∇) A
∇ • ( fA) = f (∇ • A) + A • (∇f )
∇ • ( A × B ) = B • (∇ × A) − A • (∇ × B )
∇ × ( fA) = f (∇ × A) − A × (∇f )
∇ × ( A × B ) = ( B • ∇) A − ( A • ∇) B + A(∇ • B ) − B (∇ • A)
6. Integral calculus of vectors
∫ VA .dl
C
line integral over a path l,
∫ VA .da or ∫∫VA .da
S
surface integral over a surface a,
S
∫τ Tdτ or ∫∫∫τTdτ volume integral over a volume τ.
Example
of Line integral is the work
of surface integral is flux
of volume integral is total charge in a volume
DIVERGENCE OF A VECTOR
The divergence of a vector A at a given point P is the outward
flux per unit volume:
∫ A • dS
div A = ∇ • A = lim s
∆v →0 ∆v
If the divergence of a
vector A vanishes in
space it is known to be
solenoidal.
7. (GAUSS’S) DIVERGENCE THEOREM
It states that the total outward flux of a vector field A at the closed
surface S is the same as volume integral of divergence of A.
∫ A • dS = ∫ ∇ • AdV
S V
Isn’t it clear from the definition of divergence itself???
The integral of a derivative of a function over an interval is equal to the value of
that function at the boundary or the extremities of the interval.
The boundary of a volume is a closed surface, that of a surface is a closed line.
But the boundary of a line is just two points.
CURL OF A VECTOR
The curl of vector A is an axial (rotational) vector whose magnitude is the
maximum circulation of A per unit area tends to zero and whose direction
is the normal direction of the area when the area is oriented so as to
make the circulation maximum.
A • dl
∫
Curl A = ∇ × A = lim s â n max
∆s →0 ∆s
If the curl of a vector A
vanishes in space it is
known to be irrotational.
8. STOKE’S THEOREM
The circulation 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 that A and curl of A are continuous on S.
∫ A • dl = ∫ (∇ × A ) • dS
L S
Isn’t it clear from the definition of curl it self???
