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UNIVERSIDAD NACIONAL SEDE MANIZALES
Subject: ELECTRICITY AND MAGNETISM
Workshop: FIRST PARTIAL EXAM
Teacher: LUIS FERNANDO MULCUÉ NIETO
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1 The two sides of the DNA double helix are connected by
pairs of bases (adenine, thymine, cytosine, and guanine). Because of the geometric shape of these molecules, adenine bonds with thymine and cytosine bonds with guanine. Figure shows the thymine–adenine bond.
Each charge shown is ±e. In the O-H-N combination the O- is 0.170 nm from the H+^ and 0.280 nm from the N-. In the combination the N-H-N the N-^ is 0.190 nm from the H+^ and 0.300 nm from the other N-.
(a) Calculate the net force that thymine exerts on adenine. Is it attractive or repulsive? To keep the calculations fairly simple, yet reasonable, consider only the forces due to the O-H-N and N-H-N the combinations, assuming that these two combinations are parallel to each other. Remember, however, that in the O-H-N set, the O-^ exerts a force on both the H+^ and the N-^ and likewise along the set.
(b) Calculate the force on the electron in the hydrogen atom, which is 0.0529 nm from the proton. Then compare the strength of the bonding force of the electron in hydrogen with the bonding force of the adenine–thymine molecules.
Answer: a) 8.8x10-9^ N attractive b) 8.22x10-8^ N The bonding force of the electron in the hydrogen atom is a factor of 10 larger than the bonding force of the adenine- thymine molecules.
2 Three point charges are located at the corners of an
equilateral triangle as shown in Figure.
Calculate the resultant electric force on the 7.00 μC charge. Answer: F = 0.755 i – 0.436 j N or F= 0.872 N at θ=330o
3 Four identical point charges (q = 10.0 μC) are located on
the corners of a rectangle as shown in Figure.
The dimensions of the rectangle are L = 60.0 cm and W = 15.0 cm. Calculate the magnitude and direction of the resultant electric force exerted on the charge at the lower left corner by the other three charges.
Answer: F= 40.9 N at θ=263o
4 Two point charges q 1 and q 2 are held in place 4.50 cm apart. Another point charge Q = - 1.75 μC of mass 5.00 g is initially located 3.00 cm from each of these charges and released from rest.
You observe that the initial acceleration of Q is 324 m/s^2 upward, parallel to the line connecting the two point charges. Find and q 1 and q 2
Answer: q 1 = 6.17x10-8^ C q 2 = -6.17x10-8^ C
5 Two identical beads each have a mass m and charge q.
When placed in a hemispherical bowl of radius R with frictionless, non-conducting walls, the beads move, and at equilibrium they are a distance R apart. Determine the charge on each bead.
Answer:
6 Two identical sphere with mass m are hung from silk
threads of length L as shown in Figure.
Each sphere has the same charge q 1 = q 2 = q. The radius of each sphere is very small compared to the distance between the spheres, so they may be treated as point charges. Show that if the angle is small, the equilibrium separation between the spheres is
Hint : If θ is small, then tan θ = sin θ.
7 If there is no friction and the system is in equilibrium,
determine the relationship From "Q" with "M" and with "d"
Answer:
8 A point particle of mass m and charge q is constrained
to move vertically inside a narrow, frictionless cylinder. At the bottom of the cylinder is a point charge Q having the same sign as q.
(a) Show that the particle whose mass is m will be in equilibrium at a height
(b) Show that if the particle is displaced from its equilibrium position by a small amount and released, it will exhibit simple harmonic motion with angular frequency
Hint : Take a displacement x below yo, where x<<yo, so the force can be expressed by F = -k.x. In an harmonic motion the force is F = -k.x where w^2 =k/m, the find w.
9 If there is no friction and the system is in equilibrium,
find "q" so that this state is fulfilled. (in terms of M and d).
Answer:
10 The figure shows two charges "Q" and "q" of masses
"M" and "m" in equilibrium.
a) Sketch a diagram of forces for m b) Show that the angle that the electric force with x axis is 30°. Hint : Use cosine law c) Apply ∑Fy on m to show that the tension in the rope is T=mg d) Apply ∑Fx on m to show that the charge q is
e) Sketch a diagram of forces for M f) Apply ∑Fx on M, and replace the answers of the parts c) and d). ¿What do you obtain? ¿Does it make sense?
15 A point charge is placed at each corner of a square
with side length a. The charges all have the same magnitude q. Two of the charges are positive and two are negative, as shown in Figure.
What is the direction of the net electric field at the center of the square due to the four charges, and what is its magnitude in terms of q and a?
Answer:
16 Two point charges
q 1 and q 2 are held in place 4.50 cm apart. Another point charge Q = -1.75 μC of mass 5.00 g is initially located 3. cm from each of these charges and released from rest. You observe that the initial acceleration of Q is 324 m/s^2 upward, parallel to the line connecting the two point charges. Find and q 1 and q 2
Answer: q 1 = 6.17x10-8^ C q 2 = -6.17x10-8^ C
17 Two charges are placed as shown in Figure. The
magnitude of q 1 is 3.00 μC but its sign and the value of the charge q 2 are not known. The direction of the net electric field E at point is entirely in the negative -direction.
(a) Considering the different possible signs of q 1 and q 2 , there are four possible diagrams that could represent the electric fields produced by q 1 and q 2. Sketch the four possible electric-field configurations. (b) Using the sketches from part (a) and the direction of E deduce the signs of q 1 and q 2 (c) Determine the magnitude of E
Answer:
(b) q 1 = -3.00 μC and q 2 < (c) E=1.17 x10^7 N/C
18 A charged cork ball of mass 1.00 g is suspended on a light string in the presence of a uniform electric field as shown in Figure.
When E = (3.00 i + 5.00 j ) x10^5 N/C, the ball is in equilibrium at θ=37.0°. Find
(a) The charge on the ball (b) The tension in the string.
Answer: a) 10.9 nC b) 5.44 mN
19 In the figure, find the intensity of the uniform field, so
that the sphere of charge "Q" (+) and mass "m", is in equilibrium.
Answer:
20 An electron is projected with an initial speed vo = 1.60x10^6 m/s i into the uniform field between the parallel plates in Figure.
Assume that the field between the plates is uniform and directed vertically downward, and that the field outside the
plates is zero. The electron enters the field at a point midway between the plates.
(a) If the electron just misses the upper plate as it emerges from the field, find the magnitude of the electric field. (b) Suppose that in Figure the electron is replaced by a proton with the same initial speed. Would the proton hit one of the plates? If the proton would not hit one of the plates, what would be the magnitude and direction of its vertical displacement as it exits the region between the plates? (c) Compare the paths traveled by the electron and the proton and explain the differences. (d) Discuss whether it is reasonable to ignore the effects of gravity for each particle.
Answer: (a) 364 N/C (b) The proton won’t hit the plates. The displacement is 2.73x10-6^ m, downward. (c) The displacements are in opposite directions because the electron has negative charge and the proton has positive charge. The electron and proton have the same magnitude of charge, so the force the electric field exerts has the same magnitude for each charge. But the proton has a mass larger by a factor of 1836 so its acceleration and its vertical displacement are smaller by this factor. (d) In each case a >> g and it is reasonable to ignore the effects of gravity.
21 The^ electron^ enters^ a^ region^ between^ two^ charged
plates at an angle of 37 °. Its initial speed is 5x10-6^ m/s and it is 2 cm from the positive plate. Consider the action of gravity negligible.
Determine: a) Electric field intensity. b) The time it takes to hit the plate.
Answer: a) 710.9 N/C b) q 2 = 4x10-8^ s
22 Protons are projected with an initial speed vo =
9.55x10^3 m/s into a region where a uniform electric field E = 720 N/C j is present, as shown in Figure. The protons are to hit a target that lies at a horizontal distance of 1.27 mm from the point where the protons cross the plane and enter the electric field in Figure.
Find
(a) The two projection angles ϴ that will result in a hit (b) The total time of flight (the time interval during which the proton is above the plane in Figure) for each trajectory.
Answer: a) 37° and 53° b) 167 ns and 221 ns
23 The three charges in Figure are at the vertices of an
isosceles triangle.
Calculate the electric potential at the midpoint of the base, taking q = 7.00 μC.
Answer: - 11.0 MV
24 Calculate the energy required to assemble the array of charges shown in Figure, where a = 0.200 m, b = 0.400 m, and q = 6.00 μC.
Answer: - 3.96 J
25 An Ionic Crystal. Figure shows eight point charges arranged at the corners of a cube with sides of length d. The values of the charges are +q and – q as shown. This is a model of one cell of a cubic ionic crystal. In sodium chloride (NaCl), for instance, the positive ions are Na+^ and the negative ions are Cl-
(a) Calculate the potential energy of this arrangement. (Take as zero the potential energy of the eight charges when they are infinitely far apart.) (b) In part (a), you should have found that U<0. Explain the relationship between this result and the observation that such ionic crystals exist in nature.
