Midterm Test 250S, March 7, 2012 (Use notations in the textbook); Total 30 points
Use Gauss’s law when it is appropriate
(4 pt.) 1. Consider a function v = r r nˆ.
(a) Compute the divergence of v.
(b) Check the divergence theorem ( ∇ ▯ vdτ = v ▯ da) using a sphere of radius R.
(6 pt) 2. Consider a ﬂat circular disk of radius R containing a circular hole in the middle (the
circular hole is in the region 0 ≤ s < a where s is the radial distance), and it carries a uniform
surface charge σ in the region a ≤ s ≤ R.
(a) Find the electric ﬁeld a distance z above the center of the disk.
(b) Check the limit z → ∞ (ﬁnd a ﬁrst ﬁnite term in a series expansion) Discuss the physical
implication of your limiting case (hint; rewrite the electric ﬁeld in terms of total charge instead
(6 pt) 3. Consider a hollow spherical shell that carries charge density ρ =