# Exam 1 Solutions. The ratio of forces is 1.0, as can be seen from Coulomb s law or Newton s third law.

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1 Prof. Eugene Dunnam Prof. Paul Avery Feb. 6, 007 Exam 1 Solutions 1. A charge Q 1 and a charge Q = 1000Q 1 are located 5 cm apart. The ratio of the electrostatic force on Q 1 to that on Q is: (1) none of these () 1000 (3) (4) (5) 316 The ratio of forces is 1.0, as can be seen from Coulomb s law or Newton s third law.. The charge on each of two tiny spheres is doubled while their separation distance is tripled. The ratio of the new electrostatic force to the old force is: (1) 4/9 () /3 (3) 9/4 (4) 4/3 (5) 1 Doubling the charge makes the numerator 4 times larger, while tripling the distance makes the denominator 9 times larger. Together they yield a force that is 4/9 of the original force. 3. An electron circles a carbon nucleus (6 protons, 6 neutrons) at a distance of nanometers. What is its velocity in km/s? (1) 6560 () 980 (3) 680 (4) 153 (5) 1950 Here we balance centripetal force with the coulomb force, or ( ) mv / r = k 6 e e / r, where 6e is the charge of the carbon nucleus, r is the orbital radius and m is the mass of the electron. Solving for v gives v= 6 ke / mr = m/ s= km/s. 1

2 4. The electric field required to suspend a proton against the force of gravity is: (1) 10 nanovolts/m directed upward () 10 nanovolts/m directed downward (3) 557 picovolts/m directed upward (4) 557 picovolts/m directed downward (5) none of these The E field required to balance the proton is found from Ee = mpg, where m p is the mass of the proton and e is its charge. Solving yields E = 10 nanovolts/m, directed upwards to balance the gravitational force. 5. Negative charge Q is located on the y-axis at y = a and charge Q is located at y = a. The direction of the electric field at the point (a, 0) due to these charges is toward (1) First quadrant () +y (3) x (4) +x (5) y The net y component of E is positive and the net x component is positive, putting the direction of E in the first quadrant. 6. A charge q is placed at the origin and a charge 4q is placed on the x-axis at x = cm. At what value of x on the x axis (in cm) does the electric field equal zero? (1).00 () (3) (4) 4.00 (5) 0.67 The point where E = 0 must be to the left of the charge q (why?). The x components of E from the charge q and 4q then point in the negative and positive directions, respectively. They satisfy the relation ( ) kq / x + 4 kq / x d = 0, where d is the distance between the charges. Solving for x (the negative root is the one we want) yields x = d = cm.

3 7. A thick conducting spherical shell has inner radius r and outer radius R, as shown in the diagram. A point charge of +Q is located at the center of the sphere and a charge of +6Q is placed on the conducting shell. The charge on the inner surface of the conducting shell is: (1) Q () +Q (3) +7Q (4) +6Q (5) +5Q r +Q +6Q R From Gauss law, as we showed in class, the inside charge on the outer shell must balance the charge in the center to give E = 0 inside the shell. Thus the charge on the inner shell is Q (the charge on the outer shell is +7Q). 8. An alpha particle (nucleus of helium with protons and neutrons) with kinetic energy 0 kev is shot directly at a small conducting sphere carrying charge Q and comes to a complete stop at a point 0 cm from the sphere's center. The charge Q, in microcoulombs, must be (1) +0. () (3) (4) +0.0 (5) none of these This is a problem of conservation of energy, where the initial energy is purely kinetic and the final energy is purely potential. The energy balance condition is then mv = k eq / r, where e is the charge on the alpha particle and r is the minimum distance to the sphere. Solving for Q (after converting the kinetic energy to joules) yields Q = +0. μc (must be positive to repel the alpha particle charge). 9. The electrostatic potential energy of three 10-microcoulomb charges placed at the corners of an equilateral triangle of side length 0 cm is: (1) none of these () 4.5 J (3) 9.0 J (4) 135 picoj (5) 68 J The potential energy of the system of three charges is U = 3 kq / r, where r is the distance between any two and Q is the common charge. The factor of 3 accounts for the fact that there are three pairs of charges. This yields U = 13.5 J. 1 3

4 10. An alpha particle (nucleus of helium with protons and neutrons) and an electron are each accelerated from rest through a potential difference of 500 volts. Their resultant kinetic energies, in electron volts will be, respectively: (1) none of these () 500, 1.36 (3) 10000, 500 (4) 500, 150 (5) 500, 500 The kinetic energies of the alpha particle and the electron are 5000 ev and 500 ev, respectively. 11. Electric field lines and equipotential surfaces (1) are always perpendicular to one another () make arbitrary angles when they meet (3) never cross (4) only exist for certain charge configurations (5) are always radial and circular, respectively Answer is (1), as we discussed in class. 1. The movement of charge in an electric field from one point to another at constant speed without the expenditure of work by or against the field (1) can only occur along an equipotential () can only occur along a field line (3) none of these (4) only holds when the field lines are radial (5) only occurs in a uniform field An equipotential is a surface of constant potential energy, so no work is done moving along an equipotential. 4

5 13. Two large, parallel, conducting plates are 1. mm apart and have equal and opposite charges on their facing surfaces. An electrostatic force of N acts on an electron placed anywhere between the two plates. What is the potential difference in volts between the plates? (1) 4500 () 70 (3) (4) 50 (5) The electric field is given by Ee = F, or E = F / e. The potential difference is therefore calculated from V = Ed = Fd / e = 4500 V. 14. Capacitors C 1 = 5μF and C = 10μF are connected in series across a battery of V = 100 volts and remain connected until fully charged. The battery is then carefully removed and C 1 and C are connected together, + plate to + plate and plate to plate. After charge equilibrium is reached, the potential difference across C 1 and C, respectively, is (in volts): (1) 44, 44 () 67, 67 (3) 50, 50 (4) 100, 100 (5) 0, 0 The charge on each of the capacitors in the initial configuration is found by calculating the equivalent series capacitance C s, which is 10/3 μf. The charge on C s is Q = C s V 0 = 1000/3 μc, which is the same as the charge on each of the capacitors since they are in series. When the capacitors are reconnected, charge flows from one to the other until the voltages across them equalize. In effect, the capacitors are in parallel, with the positive plates at the same potential and the negative plates at the same potential. Since the total charge does not change, we can write the total charge as ( C1+ C) V = 1000 / / 3 = 000 / 3 μc. Solving for the potential difference yields V = 000 / 45 = 44.4 V for each capacitor. 5

6 15. In the figure shown, the battery has a potential difference of 4 V. If C 1 = C = C 3 = C 4 = C 5 = 18 μf, what is the equivalent capacitance of the circuit in μf? (1) 8.8 () 11.3 (3) 18.0 (4) 1.0 (5) 36.0 C 1 + V C C 3 C 4 C 5 C and C 3 are in series and are equivalent to C 3 = 9 μf. C 3 and C 4 are in parallel and add, yielding C 34 = 7 μf. C 34 and C 5 are in series and yield C 345 = 10.8 μf. C 345 and C 1 are in parallel, yielding C 1345 = 8.8 μf. 16. A parallel-plate capacitor whose capacitance is 10 pf is charged by a battery to a potential difference of 6 V between its plates. The battery is now disconnected, and a slab of a dielectric material, whose dielectric constant is 80, is slipped between the plates. What is the stored energy of the capacitor-slab device after the slab is inserted, in joules? (1) () (3) (4) (5) When the dielectric material is inserted, the charge is the same but the electric field (and thus V = Ed) is reduced by a factor of 80 and the capacitance is increased by a factor of 80. The energy 1 CV is thus reduced by a factor of 80 from the value it would have without the dielectric. Calculating yields J. 6

7 17. Automobile and similar batteries are sometimes rated in ampere-hours. What does this represent? (1) the total charge stored in the battery () the total number of volts in the battery (3) the maximum current the battery can deliver (4) the total lifetime of the battery (5) the total energy stored in the battery Charge is current times time. 18. In recharging a weak 1-V automobile battery, it is connected to a 14-V electronically controlled charger. The initial charging current of 7.0 A decreases linearly to 3.0 A over a 6.0-hour period. At this point, how much charge (in Coulombs) has passed through the battery? (1) 108,000 () 36,000 (3) 14,400 (4) 360 (5) 40 The total charge is average current time. Average current is 5A and total time is 6 hours = 1,600 seconds, yielding a total charge of 108,000 coulombs. 19. The drift speed of electrons in a long copper wire of radius 1 mm cross section and carrying a current of 1.0 A is 0.1 mm/s. If the current is increased to 0 A, and the radius increased to mm, what would be the drift speed, in mm/s? (1) 0.5 () 1.0 (3) 4.0 (4) 0.1 (5) none of these Drift speed is proportional to current and inversely proportional to the cross sectional area (which in turn goes as the square as the diameter). This yields / = 0.5 m/s. 7

8 0. Three conductors, A, B, and C are connected across a variable voltage source. When the voltage is doubled, conductor A's current triples, conductor B's current doubles, and conductor C's current increases by Which of the conductors obeys Ohm's Law? (1) B () A, B, and C (3) A (4) A and C (5) C Ohm s law says that the current is proportional to voltage, so only B obeys Ohm s law. 8

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