PHYS 3122, Spring 2020 Homework 2

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PHYS 3122, Spring 2020

Homework 2

Policy:  On all homeworks, use whatever resources you need, including Griffiths, other texts, talking to peers, office hours, etc.  Remember that in the end, what you turn in must be your own work, reflecting your own understanding. Copying or working directly from solutions (found online, or from other students) is not considered "collaboration". Use your judgment, and remember these homeworks are intended to help you learn the material.  Please show your work or explain your reasoning whenever possible.

Due Date: Wed 01/22 by 11:59 pm (paper: in class or under my door Howey N115; electronically: Canvas).

Total:  120%, Problem C is for bonus points.

 

Exercise 1 [10%].  Let 2 be the usual separation vector from a fixed source point r/  = (x/ , y/ , z/ ) to r = (x, y, z), and let 2 be its magnitude. Show that :

the field point

(a) v(22 ) = 22,

 

(b) v(1/2) = 一 / 22 .

(c) What is the general formula for v 2n  with n ∈ N?

 

[Hint:  Realize that spatial derivatives in v = vr  are meant with respect to the field point r at fixed

r/ .]

Exercise 2 [10%]. Consider as volume ì a sphere of radius R centered at the origin. Check Green’s theorem for: (a) the vector field v1 (r) = r2  ,

(b) the vector field v2 (r) = (1/r2 ) .

(c) Comment on these results.

Exercise 3 [10%]. Write an expression for the volume charge density ρ(r) of the following charge distributions: (a) Two point charges: +3q located at r = 一D  and 一q located at r = +2D .

(b) An infinite and uniform linear charge distribution λ running along the y-axis.

(c) A spherical shell of radius R, centered at the origin, and carrying a uniform surface charge density σ .

Problem A [40%]. Consider an infinitely long cylindrical rod of radius R, axis  centered on the origin, and carrying a radially symmetric charge distribution ρ(s):

(a) Let’s first treat the case in which the charge distribution is uniform, ρ(s) = ρ0  where ρ0  is a constant.  Using Gauss’s law in integral form, calculate the E-field inside and outside the rod. [Hint:  Use a cylindrical Gaussian surface of fixed height h (along the -axis) and varying radius].

(b) What ρ(s) is required for the E-field inside the rod to have the power-law form E(s) = c sn  , where c ∈ R and n ∈ N are constants? The case n = 一1 is special, how so? Some values of n are unphysical since these would lead to an infinite amount of charge in the rod. Which values of n are physically allowed?

(c) What kind of charge distribution ρ˜(s) is required for the radial E-field inside the rod to be of constant magnitude; that is, what ρ˜(s) produces a constant E(s) (inside the rod only)?

TURN PAGE


Problem B [30%]. Liquid crystals are made of long rod-like molecules (think Cuban cigars) with a positive head and a negative tail that are packed together to form long, thin sheets.  As a function of temperature, confinement (for instance squeezing the molecules between plates) or application of an external electric field, liquid crystals organize in different phases that are qualitatively distinct from the usual phases of matter you are used to (gas, liquid and solid). One such phase, called smectic-A, is represented in the figure below. The volume charge density in the smectic phase is obviously complicated, but can be quite successfully modeled with the rather simple form ρ(z) = ρ0 sinh(z/z0 ) where ρ0  and z0  are constants, i.e. uniform in x and y, but varying in z, with z = 0 defined to be the middle of the sheet.

 

 

(a) Assuming the sheet is infinitely long in the xy-plane, use Gauss’s Law to find the electric field everywhere in space, in terms of the constants ρ0 , z0 , and the sheet thickness t.

(b) Sketch the magnitude of the resulting electric field. Where is it biggest?

 

Problem C [Bonus 20%]. The time-averaged electric potential of a neutral hydrogen atom is given by

 

V (r) =                  (1 + αr/2)

 

where q is the magnitude of the electronic charge, and α-1 = a0 /2, a0  being the Bohr radius. Find the distribution of charge (both continuous and discrete components) that generated this potential and interpret your result physically.















2022-02-14