(a) A field in some region points in the ±z direction everywhere . The magnitude (and sign, whether it points up or down) varies sinusoidally in the x direction, so the field E at location (x, y, z) is described by the following equation, with x in meters: E(x, y, z) = (1N/C) sin(2næ)ê %3D We will see at the very end of the semester that this, in fact, describes the electric field of a linearly polarized, transverse plane wave of light in vacuum. Sketch a diagram showing the field vectors in the xz plane. Make sure you show at least two cycles (i.e. include 0 < x < 2m or more). Use Gauss's law to show that there is no charge anywhere in this region (so this electric field can indeed exist in a vacuum.) (Hint: consider a rectangular Gaussian surface with one of the edges aligned with x direction and the other edges with the electric field. (b) In another region, the field points in the ±x direction everywhere and the magnitude (and sign, whether it points up or down) again varies sinusoidally in the x direction, so the field is described by the following equation: E(x, y, z) = (1N/C) sin(2mx)â. This represents a longitudinal wave. Sketch a diagram showing the field vectors in the xz plane, again making sure you show at least two cycles (i.e. include 0 < x < 2m or more). Show that in this case there must nonzero charge in some locations in this region (and so cannot represent the electric field in a vacuum) and d Screenshot ie charge density looks like.

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Question

Answer all parts of the PHYSICS problem please

(a) A field in some region points in the ±z direction everywhere. The magnitude (and sign, whether
it points up or down) varies sinusoidally in the x direction, so the field E at location (x, y, z) is
described by the following equation, with x in meters:
E(x, y, z) = (1N/C) sin(2Tx)2
We will see at the very end of the semester that this, in fact, describes the electric field of a
linearly polarized, transverse plane wave of light in vacuum.
Sketch a diagram showing the field vectors in the xz plane. Make sure you show at least two
cycles (i.e. include 0 < x < 2m or more). Use Gauss's law to show that there is no charge
anywhere in this region (so this electric field can indeed exist in a vacuum.) (Hint: consider a
rectangular Gaussian surface with one of the edges aligned with x direction and the other edges
with the electric field.
(b) In another region, the field points in the x direction everywhere and the magnitude (and sign,
whether it points up or down) again varies sinusoidally in the x direction, so the field is described
by the following equation:
E(x, y, 2) = (1N/C) sin(27x)ê.
This represents a longitudinal wave.
Sketch a diagram showing the field vectors in the xz plane, again making sure you show at least
two cycles (i.e. include 0 < x < 2m or more). Show that in this case there must nonzero charge
in some locations in this region (and so cannot represent the electric field in a vacuum) and
d Screenshot ie charge density looks like.

Expert Solution

Step 1

Gauss's law can be defined as the net electric flux enclosed in a surface is directly proportional to the electric charge enclosed. Mathematically defined as:

$ϕ_{E} = \frac{Q}{ε_{0}}$

Here, $ϕ_{E}$ is the electric flux through the closed surface, $Q$ is the electric charge and $ε_{0}$ is the permittivity.

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