Question

Consider the hemispherical shell of inner radius 3 and outer radius 7. The mass density at any point P(x,y,z) is directly proportional to the square of the distance from P to the origin.

Set up an integral in an appropriate coordinate system for the moment of inertia about the z-axis. Simplify your integrand as much as possible, but do not evaluate the integral.

Answer #1

Consider a lamina in the shape of a half annulus with inner
radius 5 and outer radius 9. The mass density at any point is
directly proportional to the distance to the origin. Compute the
moment of inertia with respect to the diameter.
Don’t leave the proportionality constant in the answer. You may,
however, leave m in your answer, so you may need to compute that as
well. What does diameter mean for a half annulus? It means the same...

A nonconducting spherical shell, with an inner radius of 7.1 cm
and an outer radius of 11.4 cm, has charge spread nonuniformly
through its volume between its inner and outer surfaces. The
volume charge density ρ is the charge per unit volume,
with the unit coulomb per cubic meter. For this shell ρ =
b/r, where r is the distance in meters from the
center of the shell and b = 3.8 μC/m2. What is
the net charge in the...

Charge is distributed throughout a spherical shell of inner
radius r1 and outer radius r2 with a volume density given by ρ = ρ0
r1/r, where ρ0 is a constant. Determine the electric field due to
this charge as a function of r, the distance from the center of the
shell.
In this problem the volume charge density ρ is not uniform; it
is a function of r (distance from the center.)

A long, cylindrical shell has an inner radius a and
outer radius b and carries a current i parallel
to the central axis. assume that within the material the current
density is uniformly distributed. Use Ampere's law to find
expressions for the magnetic field for (a) 0<r<a,
(b) a<r<b (c) r>b. Draw sketches and
carefully explain your work.

A spherical dielectric shell has inner radius r1,
outer radius r2, and dielectric constant k. A charge Q
is placed at the center of the sphere.
(a) Determine the polarization P in the dielectric shell.
(b) Find the bound volume charge density, ρb, inside
the dielectric shell.
(c) Find the bound surface charge density, σb, at r =
r1 and r = r2.

A spherical, non-conducting shell of inner radius r1 = 7 cm and
outer radius r2= 16 cm carries a total charge Q = 18 nC distributed
uniformly throughout the volume of the shell. What is the magnitude
of the electric field at a distance r = 11 cm from the center of
the shell? (k = 1/4πε0 = 8.99 × 109 N.m2/C2)

An infiinitely long solid conducting cylindrical shell of radius
a = 2.2 cm and negligible thickness is positioned with its symmetry
axis along the z-axis as shown. The shell is charged, having a
linear charge density λinner = -0.34 μC/m. Concentric
with the shell is another cylindrical conducting shell of inner
radius b = 13.8 cm, and outer radius c = 15.8 cm. This conducting
shell has a linear charge density λ outer =
0.34μC/m.
1) What is Ex(P), the...

Find the moment of inertia of a uniform hollow sphere of mass M,
inner radius r, and outer radius R > r, about an axis through
the center of mass. Consider your answer for the cases r → 0 and r
→ R. Does the result reduce correctly? Explain.

A solid insulating sphere of radius a = 5 cm is fixed at the
origin of a co-ordinate system as shown. The sphere is uniformly
charged with a charge density ρ = -244 μC/m3. Concentric with the
sphere is an uncharged spherical conducting shell of inner radius b
= 13 cm, and outer radius c = 15 cm.
1)What is Ex(P), the x-component of the electric field at point
P, located a distance d = 32 cm from the origin...

An infinitely long solid insulating cylinder of radius a = 2 cm
is positioned with its symmetry axis along the z-axis as shown. The
cylinder is uniformly charged with a charge density ρ = 27
μC/m3. Concentric with the cylinder is a cylindrical
conducting shell of inner radius b = 14.8 cm, and outer radius c =
17.8 cm. The conducting shell has a linear charge density λ =
-0.37μC/m.
1)
What is Ey(R), the y-component of the electric field...

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