Question

A cube of side 2a, centered at the origin and with the x, y, and z axes perpendicular to the faces of the cube, carries a frozen-in polarization of P = czzˆ, where c is a constant. Note that the magnitude of P is not constant – it is a function of z.

(a) Find the total bound charge inside the cube, and the total bound charge on each surface of the cube. The sum of all charges should add up to zero.

(b) Let E1 be electric field just inside the cube near the point (x, y, z) = (0, 0, a), and let E2 be the electric field just outside the cube near the same point. Find E2 − E1.

Answer #1

A cube of side L has one corner at the origin of coordinates,
and extends along the positive x, y, and z axes. Suppose the
electric field in this region is given by E = (a + b y2) j, where j
denotes the unit vector in the y direction, y denotes distance in
the y direction in meters, and a and b are constants. Find the
charge Q enclosed within the cube.

2. A slab of nonconductive material is perpendicular to the
z-axis, centered on the origin, has a total thickness of
40 mm, and isinfinite along the other two axes. It has a uniform
volumetric charge density of -108 pC/m^3. It contains a perfectly
spherical bubble with no charge or material in it. The bubble is
also centered on the origin and has a radius of 10 mm. Find
expressions for the electric field vector at all points on the
positive...

A ring of charge with radius R = 1.5 m is centered on the origin
in the x-y plane. A positive point charge is located at the
following coordinates: x = -10.1 m y = 16.8 m z = 17.1 m The point
charge and the total charge on the ring are the same, Q = +22 C.
Find the net electric field along the z-axis at z = 1.6 m.
Enet x=?
Enet y=?
Enet z=?
Thanks!!

The electrostatic potential of a certain charge distribution in
Cartesian is given by Φ(x,y,z) = V0/a3 xyz
exp[-(x+y+z)/a] , where Vo and a are constants What is the electric
field of this potential? Calculate the charge density of the source
of this field and What is the total charge in a cube of side a with
one corner at the origin and sides parallel to the axes? Write your
answer as a numerical multiple of ϵo Voa

A slab of thickness 2a, extending to infinity in x and y
directions but contained within –a < z < a, carries a
non-uniform current with current density J = J0 · z^2 /a^2 pointing
into the page where the y axis faces out of the page. Use Ampere’s
Law to find the magnetic field B inside and outside of the
slab.

The electric potential in a region is given by
V(x,y,z) = -10.0x2 + 20.0xyz + 6.0y3
a) Find the electric field that produces this potential?
b) Find the amount of charge contained within a cubic region in
space 20 cm on a side and centered at the point (10.0 cm, 10.0 cm,
10.0 cm).

Consider an infinite plane of charge with a charge density of
+10 µC/m2. Assume that this plane is on the y-and
z-axes. That is, assume it is perpendicular to the x-axis and that
it passes through the origin.
a) What would be the electric field strength and direction at
the point (5 m, 0, 0)?
The strength is___________N/C
The direction is Answer: -x, x, -y, y, -z, or z
b) What would be the electric field strength and direction at...

A thin ring of radius R in the x − y plane is centered at the
coordinate origin, and is charged with linear charge density λ
which depends on the polar angle θ as λ(θ) = λ0 sin(θ), where λ0
> 0.
(a) Sketch λ(θ) for θ ∈ [0, 2π].
(b) Before doing any calculations, sketch the E~ x and E~ y
vector components of the electric field at the coordinate origin,
as well as where (roughly) you expect the...

Let F~ (x, y, z) = x cos(x 2 + y 2 − z 2 )~i + y cos(x 2 + y 2 −
z 2 )~j − z cos(x 2 + y 2 − z 2 ) ~k be the force acting on a
particle at location (x, y, z). Under this force field, the
particle is moved from the point P = (1, 1, 1) to Q = (0, 0, √ π).
What is the work done by...

4. Find the point (x, y, z) on the sphere of radius 2 centered
at (0, 0, 0) in the first octant such that the product xyz is
maximized.

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