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Tutorial
Here are the solutions for the tutorial. All the associated python codes are available at the
end of the document.
1
2
3
→
−
→
−
4. Prove J = σ E using the fundamental postulates of electrostatics and your understanding
of electric current.
Using the definition of current,
∆I =
∆Q
∆t
∆y
∆t
= ρ∆Svy
= ρ∆S
where ρ is the volume charge density and ∆S is the differential surface area across which
current flows.
-vy
∆S
∆y
Figure 1: Ohm’s law
Through the definition of J,
J=
∆I
= ρvy
∆S
(1)
We know that the force exerted by an electric field on a electron is
F = −eE
Taking force to be the time average of momentum, we have
mvy
= −eE
τ
eτ
vy = − E
m
Combining eqn.(1) and eqn.(3), we have
J = −ρ
eτ
E
m
If we assume a uniform charge density (n) inside the conductor, we have
ne2 τ
E
m
J = σE
J=
where σ is known as the conductivity of the material. Thus we proved ohm’s law.
4
(2)
(3)
6. Find the charge distribution(spherical coordinate) that generates the following electric
field in
→
−
1
E = 2 (1 − cos 3r)âr V /m
r
Charge distribution can be found using Gauss law.
→
− →
−
∇. D = ρ
→
− →
−
ρ = 0 ∇. E
30 sin 3r
=
r2
5
6
7
10. A square conducting loop 3 cm on each side carries a current of 10 A. Calculate the
magnetic field intensity at the center of the loop.
Using Biot-Savart Law,
→
−
Id l × R
4πR3
l
= 4I
4πR2
3
= 40
4π(1.5)2
40
=
A/cm
3π
→
−
H =
Z
→
−
11. A rod of length l rotates about the z-axis with an angular velocity ω. If B = B0 âz ,
calculate the voltage induced on the conductor.
dψ
dt
d(A)
= −B
dt
Vemf = −
The area here refers to the area swept by the rod. Thus substituting that, we have
B0 l2 dθ
2 dt
1
= − B0 l 2 ω
2
Vemf = −
1
(A = l2 θ)
2
12. Region 1, described by 3x+4y ≥ 10, is free space, whereas region 2, described by 3x+4y ≤
10, is a magnetic material for which µ = 10µ0 . Assuming that the boundary between the
→
−
→
−
material and free space is current free, find B 2 , if B 1 = 0.1âx + 0.4ây + 0.2âz W b/m2
8
Use boundary conditions as given in the notes.
→
−
B 2 = −1.052âx + 1.264ây + 2âz
→
−
→
−
13. The electric field in air is given by E = rte−r−t âφ V/m (cylindrical system). Find B and
→
−
J.
Using Faraday’s law, we have
→
−
→
− →
−
∂B
∇×E =−
∂t
→
−
⇒ B = (2 − r)(1 + t)e−(r+t) âz
Using Ampere’s law, we have
→
−
→
− →
−
→
−
∂D
∇×H = J +
∂t
→
−
→
−
→
−
→
−
∂E
B
∇ × ( ) = J + 0
µ0
∂t
→
−
(1 + t)(3 − r) −(r+t)
J =
e
âφ
4π × 10−7
14. Show R =
ρl
A
and P = I 2 R from fundamental principles of electrostatics.
From Ohm’s law we know that,
I
= σE
A
J=
From electrostatics we know that E =
V
l
, thus
I
σV
=
S
l
l
V
R=
=
I
σA
ρl
=
A
where ρ is known as the resistivity of the material.
We know that power is the rate of change of energy or force times velocity. Thus
→
− −
P = F .→
u
Z
→
− −
=
ρdv E .→
u
V
Z
→
− →
=
E .ρ−
u dv
V
But we know J = ρu,
Z
P =
V
9
→
− →
−
E . J dv
If we split the volume integral as a line and surface integral, we have
Z
Z
−
→
→
− →
→
− −
=
E . dl
J .dS
=VI
⇒ P = I 2R
15. In free space, the electric field component of a TEM wave is
→
−
E = 5 sin(3 × 108 t + y)âz V /m
Determine
(a). Find wavelength, time period and wave velocity
Wavelength:
2π
k=
λ
λ = 2π m
Time period:
2π
ω
2π
=
s
3 × 108
T =
Wave velocity:
ω
k
= 3 × 108 m/s
v=
(b). Sketch the wave at t = 0, T/4, T/2
blue: t=0, green: t=T/4, red: t=T/2
10
→
−
(c). Calculate the corresponding H .
→
−
→
−
E
H = − âx
η
= −13.26 sin(3 × 108 t + y)âx mA/m
Codes
(1) from scipy import *
from pylab import *
X,Y = meshgrid(arange(-20.1,20.1,1),arange(-20.1,20.1,1))
U = -Y/(X**2+Y**2)
V = X/(X**2+Y**2)
figure(1)
Q = quiver(U,V)
figure(2)
Q = quiver(-U,V)
show()
(15) from scipy import *
from pylab import *
T = [0,pi/2,pi]
y = linspace(0,2*pi,100)
for t in T:
E = 5*sin(t+y)
plot(y,E)
show()
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