Instructions: Please provide a brief verbal explanation of each step in your solution. Statewhere the formulas are coming from, and why they are applicable here. Use symbols andformulae e?ectively de?ning their meaning and making it clear whether they are vectors orscalars. Write legibly, and draw large and clearly labeled sketches.Here is a problem that will let you both practice Gauss’s Law and help you see how morecomplicated systems can be built from the simpler ones. More complicated systems cannotbe solved by themselves with Gauss’s Law, but the simpler ones can, and then you can usethe principle of superposition to put everything together.(a) An in?nite plate with thickness 2h is parallel to the x ? z plane so that it’s mid-point(the point halfway through the plate) is at y = 0. (That way all the points on onesurface have y = +h and on the other y = ?h.) The plate is uniformly charged withvolume charge density +?. Sketch the electric ?eld lines. Using a cylinder of height2y and base area A as a Gaussian surface, ?nd the magnitude of the electric ?eld forany value of y . Graph Ey (y ) (the projection of E onto the y axis). Finally, expressˆE (x, y, z ) using the unit vectors of Cartesian coordinate system: ˆ, ˆ, and k .ij(b) An in?nite cylinder with outer radius h is coaxial with the z axis. It is uniformlycharged with the volume charge density +?. Using a cylinder of radius r and length(also coaxial with the z axis) as the Gaussian surface, derive E (r). Graph E (r).Express E (r) using the unit vector r of the vector r, drawn from the z axis to theˆpoint where we want the electric ?eld.ˆNext, express E (x, y, z ) as a function of ˆ, ˆ, and k .ij[Note: If we label the azimuthal angle of the cylindrical coordinate system with ?, thenr = xˆ + yˆ = r cos ?ˆ + r sin ?ˆ and thus r = cos ?ˆ + sin ?ˆ.](c) Through an in?nite charged plate described in part (a), an in?nitely long cylindricalhole of radius h is drilled so that it is coaxial with the z axis. [Note the cylinder axisof the hole is parallel to the plane. The system consists of a plate from part (a) withthe cylinder from part (b) taken away. ]Using the principle of superposition, and relying on the answers from parts (a) and(b), explain in words how we can get an electric ?eld at an arbitrary point.For the points along the y axis, graph Ey (y ).Write a formula for E at an arbitrary point (x, y, z ).