Institutt for fysikk, NTNU TFY4155/FY1003: Elektrisitet og magnetisme Spring 2005

Øving 2

Guidance: January 20. and 21. Deliver no later than: Monday January 24.

Exercise 1 (from earlier midterm exams)

a) A balloon has an excess of 5 · 10^{13} electrons. Then, the balloon has net charge

A 80 μC  B -80 μC  C -8 μC  D −3.2 · 10^{−33} C

b) Two point charges q and −q are located on the x axis, with q in (x, y) = (a, 0) and −q in (−a, 0). The force from these two on a third point charge q in (−a, −2a) is

\begin{matrix}A & [−x̂ + (2\sqrt{2} − 1) ŷ] F_{0}/8\sqrt{2} \\ B & [−x̂ − (2\sqrt{2} − 1) ŷ] F_{0}/8\sqrt{2} \\ C & [x̂ − (2\sqrt{2} − 1) ŷ] F_{0}/8\sqrt{2} \\ D & [x̂ + (2\sqrt{2} − 1) ŷ] F_{0}/8\sqrt{2} \end{matrix}

where F_{0} = q^{2}/4πε_{0}a^{2}.

c) Two positive and two negative point charges, all four equally large in absolute value (q) are to be placed in each of the four corners of a square. How should the four point charges be placed, in order to obtain the largest possible electric field strength on the center of the right edge, in the point P?

A 1
B 2
C 3
D 4

On a midterm exam, questions like these should be answered with a single letter only, i.e., without handing in the calculations or reasoning behind the answer. Since this is a regular home exercise, I suggest you provide both calculations and reasoning.

Exercise 2

A thin rod of length L has a uniform charge λ pr unit length.

a) What is the charge dq on a small length dx of the rod? What is the total charge Q on the rod?

b) Show that the electric field in a point P a distance R from the rod is given by E = E_{x}x̂+E_{y}ŷ, with

E_{x} = \frac{λ}{4πε_{0}R}(cos θ_{1} − cos θ_{2})
E_{y} = \frac{λ}{4πε_{0}R}(sin θ_{1} − sin θ_{2})

Here, θ_{1} and θ_{2} are the angles between the lines from P to the end points of the rod and the line passing through P being perpendicular to the rod, as shown in the figure. (The sign of the two angles are as indicated in the figure, i.e., θ_{2} is negative.)

[Hint: The field dE from a small piece dx of the rod is dE = (λ dx/4πε_{0}r^{2})r̂, where r is the vector from dx to P. Next, try to obtain an expression where θ is the integration variable, by finding the relation between x and θ.]

c) Determine the field when P is equally far from the two ends of the rod. What is E when P is very far away from the rod (i.e. R ≫ L). NB: Here, we are not interested in the trivial answer E ≃ 0 when R → ∞, but how E depends on R ”to leading order” for R ≫ L. Is your answer what you might have expected?

d) What is the electric field in a distance R from an infinitely long uniformly charged rod? (I.e.: L → ∞)

A thin, circular disk with radius R_{0} has a uniform surface charge (i.e., charge pr unit area) σ.

a) What is the charge dq on a thin ring of the disk, with radius R and width dR? What is the total charge Q on the disk?

b) Find the electric field E in a point P on the symmetry axis, a distance a from the disk. (Hint: First, find the field dE in P from a thin ring with radius R and width dR. Next, integrate from R = 0 to R = R_{0}.)

c) What is E (again: to leading order, cf question 2c) in the two limits a ≫ R_{0} and a ≪ R_{0}, i.e., far away from and very close to the disk, respectively? (In the case a ≫ R_{0}, your answer should look familiar.)

Given information: (1 + α)^{±1/2} ≃ 1 ± α/2 if α ≪ 1. (Nothing mysterious about this: These are simply the first two terms in the Taylor expansion of the functions f(α) = (1 + α)^{±1/2} around the point α = 0.)

Answer: Exercise 3b: E = \frac{σ}{2ε_{0}} (1 − _{\frac{a}{\sqrt{a^{2}+R_{0}^{2}}}} )