this question, Figure 2 (see image). Consider the electric field of a disk of radius R and surface charge density σ along the z-axis as: (see image) a) Use this expression to find the electric field of the disk very close to the disk i.e. Z << R such that the disk looks like an infinite plane with surface charge density σ.  b) Use a Gaussian cylinder (pill box) to find the electric field of the plate at this limit (Z << R such that the disk looks like an infinite plane), and compare it with your answer from part a.

Gauss’s Law With Planar Symmetry An infinite plane carries a uniform surface charge density σ. Find an expression for the electric field it produces. Hint: your Gaussian surface should be a shell with uniform cross-sectional area A, which is going through the plane. Your answer should only depend on the variables σ and ε0. A section of the plane is shown below.

A circular ring of radius R is uniformly charged with a charge q for half of its length and -q for the other half. Imagine the ring on the xy plane. Calculate the electric field at some point, P, located on the axis, say z, going through the center of the ring, O, and at a distance d from O along this axis.

A 2D annulus (thick ring) has an inner radius Ri and outer radius Ro, and charge Q non-uniformlydistributed over its surface. The 2D charge density varies with radius r by η(r)=Cr 4 for Ri ≤ r ≤ Roand η=0 for all other r. C is a constant. Answer the following in terms of the variables given above. Note: Gauss' Law will not be useful here.a) Find an expression for C such that the total charge of the annulus is Q. Include the SI units for C next to your answer.Do the units make sense?b) Draw a clear picture and use it to set up the integral to calculate the E-field at a point on the axis of the annulus (this axis is perpendicular to the plane of the annulus) a distance z from the center of the annulus. *** You must complete all the steps short of computing the integral (i.e. your eventual answer must be an integral with only ONE variable of integration and all other variables constant.) Show that your answer has the correct SI units for the electric field.