CYK\2009\PH102\Tutorial 7
Physics II
1. A linear inhomogeneous dielectric is sandwiched between the plates of a parallel plate capacitor (separation between the plates
a distance
y
= d)
charged to the charge density
K
The permittivity of the dielectric at
y
d
= 0 1 + K
where
σ.
from one of the plates, is given by
is a positive constant. (Neglect edge eects.)
(a) Find the expressions for
E, D,
and
(b) Find the bound charge densities
σb
P.
Plot these quantities as a function of
and
ρb .
y.
ρb .
Plot
(c) Find the potential dierence between the plates.
Solution
(a) First note that the
E, D and P are functions
σ . Using Gauss law,
of
y
only and are in
density on the plates be
D(y) = σ
E(y) = D(y)/(y) =
0
σ
1+K
y d
σK yd
P (y) = D − 0 E =
1 + K yd
(b) The bound surface charge density
σb (y = 0) =
σb (y = d) =
P · (−ŷ)|y=0 = 0
σK
P · (ŷ)|y=d =
1+K
The bound volume charge density
d
P
dy
σK
1
= −
d 1+K
ρb (y) = −
(c) The potential dierence
ˆ
V
= −
y 2
d
d
E dy
0
= −
The limiting value, as
K → 0,
is
−σd/0 ,
σd
ln (1 + K)
0 K
as expected.
Ε0 E
P
D
Ρb
d
d
1
ŷ
direction. Let the charge
2. A current is owing in a thick wire of radius
current density at a distance
r
a.
The current is distributed in the wire such that the
from the axis is given by
r2
1+ 2
a
J = J0
!
.
Find the total current through the wire.
Solution:
Take a cross section of the wire that is perpendicular to the axis. Then, net current is
ˆ
J · dS
I =
S
ˆ a ˆ 2π
J0
=
0
=
0
r2
1+ 2
a
!
dr(rdφ)
3
πJ0 a2
2
3. Consider a wire, bent in a shape of a parabola, kept in XY plane with focus at origin. The disntace
from apex to focus is
d.
I.
The wire carries current
Find the magnetic eld at origin.
Solution:
Equation of the parabola:
Let
r = rr̂
r (1 + cos θ) = 2d
dl =
Then,
with
θ : −π → π
be the vector pointing to the parabola. The dierntial tangent vector is given by
I dl × (0 − rr̂) = Ir2 dθẑ.
B(0) =
=
=
=
d
r dθ =
dθ
dr
dr̂
+r
dθ
dθ
dθ
dr
=
r̂ + rθ̂ dθ
dθ
r̂
Then the magnetic eld at origin
ˆ
µ0 π Ir2 ẑ
dθ
4π −π r3
ˆ π
µ0 Iẑ
1
dθ
4π −π r
ˆ
µ0 Iẑ π
(1 + cos θ) dθ
8πd −π
µ0 I
ẑ
4d
r
Θ
4. [G5.44] Use the Biot-Savart law to nd the eld inside and outside an innitely long solenoid of radius
R,
with
n
turns per unit length, carrying a steady current
and Eq 5.39. Do
z -integration
rst.]
2
I.
[Write down the surface current density
5. Consider a circular ring, of radius
R
and carrying current
I
is placed in the XY plane with its center
at origin. Set up the integral to nd the magnetic eld at a point on the X axis, at a distance
from the origin. Now, expand the integrand in the powers of
the eld. Express in terms of
m = I(πR2 ).
Solution:
3
R/d
d ( R)
and nd the rst non-zero term of
0
• r = dx̂
• r = Rŝ = R (cos φx̂ + sin φŷ)
0
1/2
0
Θ
• r − r = d2 + R2 − 2Rd cos φ
r¢
• dl = Rdφ φ̂ = Rdφ (− sin φx̂ + cos φŷ)
r
Then the magnetic eld at
0
r
• Idl × r − r = IR (R − d cos φ) dφẑ
is
0
B(r ) =
µ0
4π
ˆ
2π
IR (R − d cos φ) dφẑ
(d2
0
+ R2 − 2Rd cos φ)3/2
Now use,
"
1
3
|r0 − r|
=
#3
1
(d2 + R2 − 2Rd cos φ)1/2
"
#3
"
=
∞ n
R
1 X
Pn (cos φ)
3
d n=0 d
≈
1
R
1 + 3 cos φ + O
d3
d
R
d
2 !#
Then,
ˆ
µ0 IRẑ 2π R
R
B(r ) =
− cos φ 1 + 3 cos φ dφ
4πd2 0
d
d
ˆ 2π R
µ0 IRẑ
2
− cos φ +
≈
1 − 3 cos φ dφ
4πd2 0
d
µ0 IπR2
ẑ
= −
4πd3
0
This is consistent with the dipole eld in the direction perpendicular to the direction of the dipole.
6. [G5.6]
(a) A phonograph record carries a uniform density of static electricity
velocity
ω,
what is the surface current density
(b) A uniformly charged solid sphere, of radius
spinning at a constant angular velocity
point
(r, θ, φ)
ω
R
K
at a distance
and total charge
about the
within the sphere.
Solution:
4
r
z
σ.
If it rotates at angular
from the center?
Q,
is centered about origin and
axis. Find the current density
J
at any
7. [G5.47] Find the magnetic eld at a point
z > R
rotating sphere, of problem G5.6
5
on the axis of (a) the rotating disk and (b) the
6