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

Øving 1

Guidance: January 13. and 14. Deliver no later than: Monday January 17.

Exercise 1

a) The components of a vector A are A_{x} = 8.5 and A_{y} = −1.3. The length A = |A| of this vector is then

A) 5.6  B) 7.2  C) 8.6  D) 9.8

b) The angle between the x axis and the vector A = −3.7 x̂ + 2.3 ŷ is (in degrees, counterclockwise)

A) 32  B) 148  C) 212  D) 238

c) The components of two vectors A and B are, respectively, A_{x} = 4.1, A_{y} = −7 and B_{x} = −6.6, B_{y} = −3.1. The length of the vector B − A is then

A) 11.4  B) 14.6  C) 19.5  D) 23.3

d) The components of two vectors A and B are, respectively, A_{x} = −6.1, A_{y} = −5.8 and B_{x} = −9.8, B_{y} = 4.6. The scalar product A · B is then

A) -9.7  B) 0  C) 33.1  D) 86.5

Exercise 2

Determine the gravitational force F_{g} between two oxygen molecules (O_{2}) separated by 300 Å. Is F_{g} attractive or repulsive? The two oxygen molecules are given an extra electron each. How big is the electric force F_{e} between the two ions (O_{2}^{−})? Is F_{e} attractive or repulsive? Determine the ratio between F_{e} and F_{g}.

[Molecular oxygen has mass 32 g/mol, 1 mol = 6.02 · 10^{23}, the gravitation constant is G = 6.67 · 10^{−11} m^{3}/kg·s^{2}, e = the elementary charge = 1.6 · 10^{−19} C and 1 Å = 1 angstrom = 10^{−10} m]

Exercise 3

Two equal point charges q are positioned on the x axis in x = a and x = −a, respectively. Determine the resulting electric force F on a third point charge −q which is located on the y axis (y > 0) in a distance \sqrt{5} a from the two others. Draw a figure with the three charges where you clearly indicate the direction of F. [Notation: vectors with bold letters.] Determine a numerical value of F when q = 2 μC and a = 4 cm. What is the force on the charge on the y axis if the point charge in x = −a changes sign? (Draw a figure.)

Exercise 4

a) Six identical charges q are located in the corners of a regular hexagon. What is the force on a test charge Q in the centre of this hexagon?

b) One of the six charges are removed. What is the resulting force on Q? Draw a figure and explain your reasoning.

c) Replace ”six” with ”seven” and repeat question a!

Exercise 5

a) Two point charges 3q and −q are located on the x axis in x = 0 and x = a, respectively. Explain why possible equilibrium positions for a third charge q must be on the x axis.

b) There is one equilibrium position x_{0} on the x axis for this third charge. (In addition to the ”singular” point x = a.) Determine x_{0}. Argue, without further calculations, that this equilibrium position is unstable with respect to a small displacement in the x direction. (Alternatively, with further calculations: Consider the stability of the equilibrium by looking at the value of dF/dx in x = x_{0}.)

[In equilibrium, there is no net force on the charge. If the charge is displaced a distance Δx from the equilibrium position, it will be influenced by a force. If this force acts in the same direction as the displacement Δx, the equilibrium is unstable. In the opposite case, it is stable.]

Some answers:

Exercise 2: F_{e}/F_{g} ≃ 10^{33} Exercise 3: F = 8.0 N. Exercise 5b: x_{0} = (3 + \sqrt{3})a/2.