Instructions: 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: ˆ, ˆ, 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
(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 ˆ, ˆ, and k .
[Note: If we label the azimuthal angle of the cylindrical coordinate system with φ, then
r = xˆ + yˆ = r cos φˆ + r sin φˆ and thus r = cos φˆ + sin φˆ.]
(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 ).