EET 303: ELECTROMAGNETIC THEORY
SEM 1 2018/2019
TUTORIAL 1: VECTOR ANALYSIS
1.
At point P(− 3,4,5), express the vector that extends from P to Q(2,0,−1) in:
(a) Cartesian coordinates.
(b) Cylindrical coordinates
(c) Spherical coordinates
(d) Show that each of these vectors has the same magnitude
2.
Find the gradient of each following:
(a) V1 = V0 e−2r sin 3ϕ
(b) V2 = V0 (a )cos2θ
R
!
3.
Given A = r cosφa r + r sin φa φ + 3za z , find ∇ • A at (2,0,3).
4.
Evaluate the line integral of
E = xa x − ya y along the segment P1 to P2 of the
circular path shown in the Figure 1.
Figure 1
5.
Transform the vector A = 3a x − 2a y + za z into spherical vector and evaluate it at
the point P(2, − 1, 1) .
!
6.
Find ∇ × A at (3, π/6, 0) for the vector field
!
A = 12 sin θ aθ
7.
For the vector field, E = 10e
−r
a − 3z a , verify the divergence theorem for the
r
z
cylindrical region enclosed by r =2, φ =0 and z = 4.
8.
(
)
ˆ = r cosφ + sin φ
Verify Stokes’s theorem for the vector field B
ar
aφ by evaluating:
(a) ∫ B • dl over the semicircular contour shown in Figure 2, and
C
(b)
∫ (∇ × B) • dS over the surface of the semicircle.
S
Figure 2