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# final exam 2017J

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```center due to a homogeneously charged infinite thick slab (thickness
linear with z, within the slab, and constant outside (12 N/C). The field is
to the slab.
ELECTRONICS AUTOMATION ENGINEERING
a) Determine the charge density in the slab.
MECHANICAL ENGINEERING
PHYSICS II
JUNE 2017
b) Determine the electric field at (0,0,a) after having hollowed out a sph
FINAL
radius a/2, centered at the coordinate
origin.EXAM
1. The figure shows the plot of the electric
field z-component, Ez , versus z (left) due
to a homogeneously charged infinite thick
slab (right), where thickness 2a = 10 cm.
Ez is linear with z within the slab, and
constant outside (12 N/C). The field is
perpendicular to the slab. (a) Determine
the charge density in the slab. (b) Determine the electric field at point (0, 0, a) after having hollowed out a spherical cavity
of radius a/2 centered at the origin of coordinates. (c) Calculate the electric field
at point (0, 0, a/2) if the charge density in the slab is given by the expression ρ(z) = Az 2 ,
where A = 2.5 &times; 10−6 C/m5 .
2. A hollow metal spherical shell of radius R1 = R is charged to a potential V . (a) Calculate
the electric field everywhere. (b) Repeat part (a) if, maintaining the previous potential
connected, the shell is surrounded by another concentric shell of radius R2 = 2R connected
to the ground. (c) For the case of part (b), what would be the maximum potential V at
which the inner shell could be connected before air breaks down if R = 10 cm? Assume
that air has a breakdown field strength of 3 MV/m and dielectric constant of 1, and that
the potential at infinity is zero, the same as ground.
y
3. The figure shows the cross section of radius 2R of an infinitely long cylindrical cable. The cable carries two currents I going in opposite directions. One goes in the positive z direction and is uniformly distributed over a circle of
radius R, and the other goes in the negative z direction
I and 2R
is uniformly distributed over the rest of the cross section.
O
Calculate the magnetic field B (a) at points x &gt; 2R,Ry = 0 I
and (b) at points 0 &lt; x &lt; 2R, y = 0.
4. Two infinite parallel straight wires carrying a current I in opposite directions, and a rectangular coil
of sides a and b are in the same plane, as shown
in the Figure. Find: (a) The magnetic field B as
a function of y inside the coil. (b) The magnetic
flux through the coil. (c) The induced e.m.f. in
the coil when the current I varies in time as
t ≤ 0, I = 0; 0 ≤ t ≤ T, I = αt; t ≥ T, I = αT.
y
I
2R
x
x
O
R
I
y
y
I
c
b
I
I
a
c
c
a
b
c
I
PHYSICAL CONSTANTS
e = 1.6 &times; 10−19 C
K = 9 &times; 109 N &middot; m2 /C2
ε0 = 8.85 &times; 10−12 C2 /N &middot; m2
&micro;0 = 4π &times; 10−7 N/A2
ELECTRICITY
F = qE
I
ΦE =
C=
Q
V
Qenclosed
ε0
1
1
U = QV
u = εE 2
2
2
E &middot; dA =
MAGNETISM
F = qv &times; B
&micro; = N IA
dF = Idl &times; B
τ =&micro;&times;B
&micro;0 Idl &times; ur
dB =
4π
r2
I
B &middot; dl = &micro;0 Ienc
I
B = &micro;0 (H + M) = &micro;H
H &middot; dl = If
Z
dΦB
ΦB = B &middot; dA
E =−
dt
E = −L
dI
dt
1
U = LI 2
2
u=
1 B2
2 &micro;
```