Please provide a brief verbal explanation of each step in your solution. State where the formulas are coming from, and why they are applicable here. Use symbols and formulae effectively defining their meaning and making it clear whether they are vectors or scalars. 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 more complicated systems can be built from the simpler ones. More complicated systems cannot be solved by themselves with Gauss's Law, but the simpler ones can, and then you can use the principle of superposition to put everything together.
(a) An infinite 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 one surface have y = +h and on the other y = -h.) The plate is uniformly charged with volume charge density +ρ . Sketch the electric field lines. Using a cylinder of height 2y and base area A as a Gaussian surface, find the magnitude of the electric field for any value of y. Graph Ey (y) (the projection of E→ onto the y axis). Finally, express E→ (x, y, z) using the unit vectors of Cartesian coordinate system: i^, j^, and k^.
(b) An infinite cylinder with outer radius h is coaxial with the z axis. It is uniformly charged with the volume charge density +ρ. Using a cylinder of radius r and length A (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 the point where we want the electric field.
Next, express E→ (x, y, z) as a function of i^, j^, and k^.
[Note: If we label the azimuthal angle of the cylindrical coordinate system with Φ, then r→ = xiˆ + yjˆ = rcos Φiˆ + rsin Φjˆ and thus rˆ = cos Φiˆ + sin Φjˆ.]
(c) Through an infinite charged plate described in part (a), an infinitely long cylindrical hole of radius h is drilled so that it is coaxial with the z axis. [Note the cylinder axis of the hole is parallel to the plane. The system consists of a plate from part (a) with the 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 field 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).