Download EE3321 ELECTROMAGENTIC FIELD THEORY

Week 2
Vector Operators
Divergence and Stoke’s Theorems
Gradient Operator
 The gradient is a vector operator denoted  and sometimes also
called “del.” It is most often applied to a real function of three
variables.
 In Cartesian coordinates, the gradient of f(x, y, z) is given
by
grad (f) =  f = x ∂f/∂x + y ∂f/∂ + z ∂f/∂z
 The expression for the gradient in cylindrical and spherical
coordinates can be found on the inside back cover of your
textbook .
Significance of Gradient
 The direction of grad(f) is the
orientation in which the
directional derivative has the
largest value and |grad(f)| is
the value of that directional
derivative.
 Furthermore, if grad(f) ≠ 0,
then the gradient is
perpendicular to the “level”
curve z = f(x,y)
Example
 As an example consider the gravitational potential on the
surface of the Earth:
V(z) = -gz
where z is the height
 The gradient of V would be
 V = z ∂V/∂z = -g az
Exercise
 Consider the gradient represented by the field of blue
arrows. Draw level curves normal to the field.
Exercise
 Calculate the gradient of
 f = x2 + y2
 f = 2xy
 f = ex sin y
Exercise
 Consider the surface z2 = 4(x2 + y2). Find a unit vector
that is normal to the surface at P:(1, 0, 2).
Laplacian Operator
 The Laplacian of a scalar function f(x, y , z) is a scalar
differential operator defined by
 2 f = [∂ 2 /∂x 2 + ∂ 2 /∂y 2 + ∂ 2 /∂z 2 ]f
 The expression for the Laplacian operator in cylindrical
and spherical coordinates can be found in the back cover of
your textbook .
 The Laplacian of a vector A is a vector.
Applications
 The Laplacian quite important in electromagnetic
field theory:
 It appears in Laplace's equation
2 f = 0
 the Helmholtz differential equation
 2 f + k2 f = 0
 and the wave equation
2 f = (1/c)2 ∂2 f/∂x2
Exercise
 Calculate the Laplacian of:
 f = sin 0.1πx
 f = xyz
 f = cos( kxx ) cos( kyy ) sin( kzz )
Curl Operator
 The curl is a vector operator
that describes the rotation of a
vector field F:
xF
 At every point in the field, the
curl is represented by a vector.
 The direction of the curl is the
axis of rotation, as determined
by the right-hand rule.
 The magnitude of the curl is
the magnitude of rotation.
Definition of Curl
where the right side is a line integral around an
infinitesimal region of area A that is allowed to shrink
to zero via a limiting process and n is the unit normal
vector to this region.
Line Integral
 A line integral is an integral where the function is
evaluated along a predetermined curve.
Significance of Curl
 The physical significance of
the curl of a vector field is
the amount of "rotation" or
angular momentum of the
contents of given region of
space.
Exercise
 Consider the field shown
here.
 If we stick a paddle wheel in
the first quadrant would it
rotate?
 If so, in which direction?
Curl in Cartesian Coordinates
 In practice, the curl is computed as
 The expression for the curl in cylindrical and spherical
coordinates can be found on the inside back cover of your
textbook .
Exercise
 Find the curl of F = x ax + yz ay – (x2 + z2) az.
Divergence Operator
 The divergence is a vector
operator that describes the
extent to which there is more
“flux” exiting an infinitesimal
region of space than entering
it:
·F
 At every point in the field,
the divergence is represented
by a scalar.
Definition of Divergence
where the surface integral is over a closed infinitesimal
boundary surface A surrounding a volume element V,
which is taken to size zero using a limiting process.
Surface Integral
 It’s the integral of a function f(x,y,z) taken over a
surface.
Example
 Consider a field F = Fo/r2 ar. Show that the ratio of the
flux coming out of a spherical surface of radius r=a to
the volume of the same sphere is
= 3Fo/4a3
 First calculate
 Then calculate
= 4 π Fo
V = 4π a3/3
Significance of Divergence
 The divergence of a field is the extent to which the
vector field flow behaves like a source at a given point.
Divergence in Cartesian Coordinates
 In practice the divergence is computed as
 The expression for the divergence in cylindrical and
spherical coordinates can be found on the inside back
cover of your textbook .
Exercise
 Determine the following:
 divergence of F = 2x ax + 2y ay.
 divergence of the curl of F = 2x ax + 2y ay.
Divergence Theorem
 The volume integral of the divergence of F is equal to
the flux coming out of the surface A enclosing the
selected volume V :
 The divergence theorem transforms the volume
integral of the divergence into a surface integral of the
net outward flux through a closed surface surrounding
the volume.
Example
 Consider the “finite volume”
electric charge shown here.
 The divergence theorem can
be used to calculate the net
flux outward and the
amount of charge in the
volume.
 Requirement: the field
must be continuous in
the volume enclosed by
the surface considered.
Exercise
 Consider a spherical surface of radius r = b and a field
F = (r/3) ar.
 Show that the divergence of F is 1.
 Show that the volume integral of the divergence is
(4π/3) b3
 Show that the flux of F coming out of the spherical
surface is (4π/3) b3
Stokes' Theorem
 It states that the area integral of the curl of F over a
surface A is equal to the closed line integral of F over
the path C that encloses A:
 Stoke’s Theorem transforms the circulation of the field
into a line integral of the field over the contour that
bounds the surface.
Significance of Stoke’s Theorem
 The integral is a sum of
circulation differentials.
 The circulation
differential is defined as
the dot product of the
curl and the surface area
differential over which it
is measured.
Exercise
 Consider the rectangular surface shown below.
 Let F = y ax + x ay. Verify Stoke’s Theorem.
B
A
Homework
 Read book sections 3-3, 3-4, 3-5, 3-6, and 3-7.
 Solve end-of-chapter problems
 3.32, 3.35, 3.49, 3.39, 3.41, 3.43, 3.45, 3.48