Program: B.Tech (EE)
Semester: IV
Subject Name: Engineering Electromagnetics (EL408)
Lesson Plan – 5
Lecture Time: 60Mins.
Topic Name: Gradient of a scalar, divergence of a vector and divergence theorem.
Faculty: Kantipudi MVV Prasad
Session description:
This session provides students with knowledge of gradient of a scalar field, divergence of a vector field and
divergence theorem.
Session learning Outcomes:
The students will be able to,
•
Understand the gradient of a scalar field.
•
Express the gradient of scalar in coordinates systems.
•
Find the gradient of the scalar fields
•
Understand the divergence of a vector field at positive and negative.
•
Express the divergence of a vector at any point in coordinates systems.
•
Determine the divergence of the vector fields
Background Information
Gradient of a scalar
“The gradient of a scalar field V is a vector that represents both the magnitude and the direction of the maximum space
rate of increase of V “
It depends upon the position where the gradient is to be evaluated and it may have different magnitudes and directions at
different locations in space.
The gradient of V can be expressed in Cartesian, cylindrical, and spherical coordinates.
Program: B.Tech (EE)
Semester: IV
Subject Name: Engineering Electromagnetics (EL408)
Divergence of a vector and divergence theorem
“The divergence of A at a given point P is the outward flux per unit volume as the volume shrinks about P”
Where
is the volume enclosed by the closed surface S in which P is located. Physically, we may regard the divergence
of the vector field A at a given point as a measure of how much the field diverges or emanates from that point. Figure (a)
shows that the divergence of a vector field at point P is positive because the vector diverges (or spreads out) at P. In Figure
(b) a vector field has negative divergence (or convergence) at P, and in Figure(c) a vector field has zero divergence at P.
Figure: Illustration of the divergence of a vector field at P; (a) positive divergence, (b) negative
divergence, (c) zero divergence.
Program: B.Tech (EE)
Semester: IV
Subject Name: Engineering Electromagnetics (EL408)
“This is called the divergence theorem, otherwise known as the Gauss-Ostrogradsky theorem”
Statement: 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.
Figure: Volume v enclosed by surface S.
Materials required:
• Black board, Chock and duster and A4 sheets need to be distributed to students for Activity
Preparatory steps:
• Read the back ground information
• Refer lecture notes and Practice drawing it yourself before the students draw.
• Review the lesson plan.
Program: B.Tech (EE)
Semester: IV
Subject Name: Engineering Electromagnetics (EL408)
Detailed Lesson Plan:
Time
20 Minutes
Activities
Resources
Other relevant
Information
Power point
presentation.
This is Teacher
centric activity,
which engages
student in learning
Will discuss Divergence of a vector and Power point
divergence theorem by considering
presentation.
This is Teacher
centric activity,
which engages
student in learning.
Gradient of a scalar field
Topic outline will be discussed for initial
05 Minutes.
Black board
Will discuss gradient concept for the
following Coordinate Systems
(i) Cartesian
(ii) Cylindrical
(iii) Spherical
With one good example, which covers the
above three Coordinate Systems.
20 Minutes
a) Positive divergence
Black board
(b) Negative divergence
(c) Zero divergence
And express the divergence expressions
for the following three Coordinate
Systems.
(i) Cartesian
(ii) Cylindrical
(iii) Spherical
10 Minutes
Clarification Pauses:
A4 sheets for
activity and
“Throughout a lecture, particularly after Power point
stating an important point or defining a presentation
key concept, stop, let it sink in, and then for display of
instructor
This is student
centric activity,
which engages
student interest in
the topic.
Program: B.Tech (EE)
Semester: IV
Subject Name: Engineering Electromagnetics (EL408)
(after waiting a bit!) ask if anyone needs program.
to have it clarified. You can also move
around the room during these pauses to
look at student notes, answer questions,
etc. Students who would never ask a
question in front of the whole class will
ask questions during a clarification pause
as you move around the room.
One Minute Paper - Ask students to take
out a blank sheet of paper, pose a
question (either specific or open-ended),
and give them one (or perhaps two - but
not many more) minute(s) to respond.
This tells you whether or not the students
are viewing the material in the way you
envisioned.
05 Minutes
Lesson closure and Assessment.
Summarize the ideas in the lesson by
revisiting the understanding Goals.
Power point
presentation.
Black board
Summarize the
lesson, clear any
misconceptions,
and end with an
interesting
question to ponder
for the next lesson.
Attendance: (05 Mins)
Reference Readings: 1. Matthew N.O. Sadiku, “Elements of Electromagnetics” Oxford Univ. Press, 4 th ed., 2007, ISBN: 0-19806229-X
2. William H. Hayt Jr. and John A. Buck, “Engineering Electromagnetics”, TATA McGraw-Hill, 7th Edition,
2006, ISBN- 10: 0-07-061223-4
3. Kraus/Fleisch, “Electromagnetics with Applications”, TATA McGraw-Hill, 5th Edition, ISBN: 0-07-116429-4.