# Physics 2220 Fall 2010 George Williams SECOND MIDTERM - REVIEW PROBLEMS

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1 Physics 0 Fall 010 George Williams SECOND MIDTERM - REVIEW PROBLEMS The last four problems are from last years second midterm. Solutions are available on the class web site.. There are no solutions for, -4, 69, 111, 115, 141, 144. Gauss' Law problems, involving electric fields, from the last review set are still fair game. Any problem involving gravity in the solution set is off limits. 1. Given an electric potential of the form V = Bx y z, where B is a constant. Calculate the y component of the electric field. Calculate the equivalent capacitance between A and B of the network shown. All capacitances are.00 pf. A.0 pf capacitor is connected in series with a.00 pf capacitor. What is the equivalent capacitance of the combination? -6 If a long metal cylinder of radius 1.00 mm has linear charge density of.5 10 C/m, what is the electric field at its surface? A 0 pf capacitor is charged to 150 V. The battery is disconnected. It is then connected in parallel to a 40 pf capacitor. How much energy is lost?. -10 Calculate the electric force between two electrons m apart. The electric potential in a region of space is given by V = -Ax y, where A is a constant. Calculate the y component of the electric field. Calculate the equivalent capacitance between A and B of the combination below. C 1 = 4.0 pf; C = 7.0 pf; C = 15.0 pf In a hydrogen atom, one electron orbits about one proton. If you assume a circular orbit (classical -10 model), and a distance between proton and electron of m, calculate the velocity of the electron. The energy stored in a capacitor is 1.75 J. If the voltage is 175 V, find the value of the capacitance.. Calculate the equivalent capacitance of the series combination of 100 pf and 1800 pf. A 1 volt battery is connected across a 100 f capacitor. Find the charge on the capacitor. Find the value of g 1000 miles below the surface of the earth. Find the electric field 1. m away from a long cylinder of radius 0.10 m. The cylinder has a surface -6 charge density of C/m. Find the electric field between the plates of a parallel plate capacitor if the area of the plates is 0.10 m, the separation is 1. mm, the voltage across the plates is 10 V, and there is not dielectric between the plates. 4. Find the effective capacitance between points A and B of the combination of capacitors shown. An electron at rest is accelerated through a potential difference of 17 V. Find its kinetic energy (in Joules).

2 Convert 750 electron-volts to Joules. If the electric potential is given by V = Ax where A is a constant, find the x component of the electric field. Find the y component of the field due to the potential in. 5. Calculate the potential energy of the square configuration of electric charges given below. Find the potential at A before the charge is placed at A, but after the other three charges are in place. Find the work to bring a charge of +q from infinity to point B at the center of the square (with all 4 charges at the corners in place). 6. Given a spherical nonconductor with a uniform charge distribution of density. Calculate the electric field at a point R o/, where R o is the radius of the sphere. Calculate the total energy stored in the electric field between R / and R. 7. Given two thin concentric conducting spheres. One has a radius of R 1 and the other a radius of R. There is a charge +Q on the inner and -Q on the outer sphere. 1 Find the electric field at a point P at R = / R 1. Find the total energy stored in the electric field between R = / R and R = R. o o Given two concentric, metal cylinders of length L and radii a and b. They are charged with charges +Q and -Q as shown. Calculate the energy density at an arbitrary point P, a distance R from the center. Set up, but do not evaluate, an integral by which one could calculate the total energy stored in this system. 9. Calculate the effective capacitance between and of the network given. All capacitances are.00 pf. 4 7 Given an electric potential of the form V = Ax y z where A is a constant. Calculate the y component of the electric field. Three charges of equal magnitude Q = 4.65 C, are the corners of an equilateral triangle whose sides a = 4.5 cm. Calculate the electric potential, including sign, at point P, at the midpoint between the two positive charges.

3 A.75 pf capacitor is charged to 50 V. Calculate the energy stored in the capacitor. Calculate the capacitance of a parallel plate capacitor if the area of each plate is.75 cm, and the plate separation is mm. The plates have a vacuum between them. 10. Calculate the electric potential (in volts) at the center of a quarter circle of -9 charged wire if the charge density is = C/m and the radius is 15.0 cm. If the potential is in the form V = Ay z, calculate the x component of the electric field at the point (1,,). 5 Calculate the electric potential difference needed to give an electron a velocity of m/s. -6 An electron is released from rest in a uniform electric field of V/m. Calculate its velocity after 1.5 seconds have passed In moving a charge of q = C from A to B, J of work is done. Calculate the potential difference between A and B. 11. Given a semi-infinite line of charge of density + c/m. The line begins at x = 0, and extends to infinity along the x axis. Calculate the electric potential difference between points A and B (V A - V B). (Hint: Instead of infinity, use a large value of x, and see what happens.) 1. The electric potential difference between two charged metal plates that are parallel is 17.0 V. Calculate the magnitude of the work to move an external small charge of 7.5 C from one plate to another. (The charge is moved slowly.) A capacitor of C = 17. pf is given a potential of 75.0 V. Calculate the charge on the capacitor. If the potential in a region of space is given by V = +A xy z volts, where A is a positive constant, calculate the y component of the electric field at the point x = +.00 m, y = +.00 m, z = +.00 m. If all capacitors in the arrangement shown have the same value C, calculate the effective capacitance between a and b. Calculate the capacitance of a parallel plate capacitor if the plates are squares with sides of m, and the gap between plates is 0.75 mm. 1. Given the configuration shown. Q 1 = C; Q = C Q = 5.00 C; a = m Calculate the electric potential, including sign, at point P. Find the work done by someone who brings a charge of C from infinity to point P. Calculate the potential energy of the configuration of 4 charges after the charge in is brought to point P.

4 14. In the arrangement shown, switch A is closed for a while and then opened. After A is open switch B is closed. Find the potential on each capacitor after B is closed. Calculate the charge on each capacitor after B is closed. V batt = 175 V; C 1 = 1.5 F; C =.40 F; C =.00 F 15. Calculate the electric potential at point P due to the four charges shown. Calculate the electric potential at point S due to the four charges shown. 16. A long cylindrical rod (non-conductor) has a uniform density of C/m. The cross section of the rod is circular with a radius of R. Calculate the electric field at an arbitrary interior point of the cylinder, a distance R A from the center, and far from either end. If the rod is negatively charged, state the sign of V(R o) - V(R A). Explain clearly in words why this is the correct answer Given an electric potential of the form V = Ax y z, calculate the y-component of the electric field at the point x = +.00, y = -.00, z = A is a constant. Find the electric field at a point 7.5 cm from the center of a conducting sphere. The sphere has a charge of C and a diameter of 0.0 cm. Given a non-conducting sphere of radius 5.00 cm whose total charge (uniformly distributed) is C. Find the work needed from the external world to move one electron from R = 7.00 cm to R = 11.5 cm. R is measured from the center of the sphere. 18. Calculate the work necessary to assemble the charge configuration shown. Find the work needed to bring a charge Q from infinity to point P. P is halfway between the two charges. o 19. Consider a long cylindrical non-conductor. The outside radius of the cylinder is R o, and the charge distribution inside the cylinder is modeled by = AR, where A is a constant and R R (artificial, but it keeps the math simple). o Calculate the electric field at any interior point at a distance R from the center of the cylinder. Using V(R) = 0 at R = 0, find the value of the potential at a point R where R R o. The sign of V must be clearly stated for the case where the charge density is positive. Obtain a formula for A if the cylinder has a linear charge density of C/m. What are the units of A?

5 Calculate the work needed to assemble protons (q = C) into an equilateral triangle m on a side. If 15.0 coulomb of charge is uniformly distributed along a wire.00 km long, calculate the electric field at the mid point of the wire and 1.50 cm from the center of the wire and perpendicular to the wire. 6 Calculate the term in a for the expression given below. a << x. An electric potential is given by V = Axy (1 - z ). Find the y component of the electric field at the point x = +1.00, y = +.00, z = In the geometry shown the charge +Q is at x = +a and the charge -Q is at x = +a. The point P is at x = 0, y = +a. Find the electric potential at point P (with the standard convention for V = 0). If Q = 750 pc and a = 1.5 cm, calculate a numerical value for the potential at P, including the sign. How much work (numerical answer) is required to bring an electron from infinity to point P?. If the electric potential in a region of space is calculated as V = Ax y z, calculate the z component of the electric field. -/ For the function (1 - x), calculate a numerical result for the coefficient of the term in x (assume x < 1). -5 A long rod of non-conductor has a uniform linear charge density given by = C/m. Calculate the magnitude of the electric field.0 cm away from the center of the rod (the radius of the *rod is <.0 cm). The electric field near the surface of a conductor is found to be +7 V/m. Calculate the surface charge density on the conductor at the point where this electric field is measured. An electron is accelerated from rest by a potential difference of 10.0 volts. Calculate its velocity.. For the configuration of charges shown, calculate the potential at point P (numerical value) with the standard convention for V = 0. How much work is required (with sign) to bring an electron from infinity to point P? Q = C; Q = C; Q = C; a = 1.7 cm

6 4. A long hollow tube of nonconductor has an inside radius of R A and an outside radius of R B. The nonconductor has a uniform charge density (C/m ). The charge density is zero everywhere else. State in words the shape of the Gaussian surface that you use in each case. Calculate the electric field for R < R A. Find the electric field for R A < R < R B (in the interior of the nonconductor). Calculate the electric field for R > R B (outside the tube, assuming R is less than the length of the tube). 5. Determine the work, in Joules, to bring an electron and proton from far apart to a distance of m apart. Calculate the equivalent capacitance between A and B for the network shown. All capacitances are.00 pf. 6 A F capacitor is charged to 150 volts. Find the energy stored in this capacitor. The electric potential in a region of space is given by V = Ax y z, where A is a constant. Calculate the magnitude of the z component of the electric field at the point x =, y =, z = 4. [All quantities in appropriate SI units.] A parallel plate capacitor has square plates m on a side that are m apart. If this 4 capacitor stores.5 x 10 J of energy, find the electric field between the plates. There is no dielectric between the plates. 6. A square distribution of charges is set up as shown. The side of the square is "a." Calculate the electric potential, in volts, at point C, the center of the square. Calculate the electric potential, in volts, at point D, the middle of the bottom side of the square. Find the work, in Joules, necessary to bring a charge Q = C to point C, from infinity. Q = C Q = C a = m 1 4 Q = C Q =.5 10 C 7. For the arrangement shown, calculate the equivalent capacitance between A and B. If the battery is 150 V, what is the potential across C 1? If the battery is 150 V, what is the charge on C? C = 1.00 pf C =.00 pf C = 5.00 pf C =.00 pf C = 4.00 pf 5

7 8. A capacitor is charged to 5 volts. The value of the capacitance is 600 pf. Calculate the energy stored in this capacitor. 4 If the electric potential in region of space is given by v = Bx y z, where B is the constant, calculate the y component of the electric field (including sign) at the point x =, y =, z = 4. A parallel plate capacitor has dimensions of 4.00 cm by 15 cm. The plates are 0.15 mm apart and there is a dielectric with = 4.5 between the plates. Calculate the capacitance of this capacitor. If the earth (use R = 6000 km) is found to have a potential of exactly 1.00 million volts, what is the charge on the earth? Each of the capacitors in the network shown has the same value C. Calculate the equivalent capacitance between A and B. 9. A battery of 150 V is connected between points A and B. Calculate the equivalent capacitance between points A and B. Calculate the charge and potential on capacitor C. C 1 = 60.0 pf; C = 45.0 pf; C = 5.0 pf; C 4 = 5.0 pf 0. All capacitors have the same value, C. Calculate the equivalent capacitance between a and b. A.50 pf capacitor is charged to 150 volts. Calculate the energy stored in this capacitor in joules. A positive charge, +Q, and a negative charge, - Q, are a distance a apart. Calculate the electric potential half-way between them, at point P. 17 If bismuth has charge carriers/cm, calculate the drift velocity, in m/s, in a square wire 1.00 mm on a side carrying a current of 4.50 A. A 1.50 F capacitor is charged to 00 volts and disconnected from the power source. Now a dielectric of =.00 is inserted fully in the capacitor. Calculate the change in the stored energy in the capacitor and state whether it is an increase or decrease. 1. For the network shown, the switch S is closed long enough for the capacitors to be fully charged, and then it is opened. Calculate the potential and charge for each capacitor. Now, with the switch still open, a dielectric with = 4.00 is inserted into C. Calculate the new values of charge and potential for each capacitor. C 1 = 1.0 F; C = 1.50 F; C = 0.50 F; =.00 V

8 . 5 Given an electric potential of the from V = Ax y z, calculate the magnitude of the electric field at the point x =, y = 4, z = 6. 4 Given the function, calculate the coefficient of the term in x in the binomial expansion. The electric field just above the surface of a sphere of radius 5.00 cm is 150 N/C. Calculate the total charge on the sphere. -19 In moving an electron from point A to point B J of work is done from the outside world. Calculate the potential difference V B - V A, with sign. If all capacitors have the same value, C, calculate the effective capacitance between a and b.. The points shown are all along the x-axis. Determine the electric field, magnitude and direction, at point A. Calculate the electric potential at point A using the usual assumption about V = 0. Calculate the electric field, magnitude and direction, at point B. Find the electric potential at point B. 4. The capacitor system shown is connected to a battery with V B = 180 V. Calculate the equivalent capacitance between points A and B. Calculate the potential across C 6. Calculate the charge on C C = 1.00 F C =.00 F C =.00 F C = 4.00 F C = 5.00 F C = 6.00 F 5. A very long cylinder of non-conductor and radius R 0 is charged with a volume charge density given by = BR between R = 0 and R = R o. B is a constant. Calculate the magnitude of the electric field at any arbitrary distance R from the center (for R < R o). Calculate the magnitude of the potential difference between R = 0 and R = R o. If the rod is negatively charged, what is the sign of V(R o) - V(0)? Calculate the energy stored in the electric field for 1 meter length of this cylinder between R = 0 and R = R /. o

9 6. With the switch S closed, a dielectric of =.50 is inserted into C. Then the switch is opened. After opening the switch the dielectric is removed from C. At this point, with the switch still open, calculate: the voltage and charge on C 1, the voltage and charge on C the voltage and charge on C, and the voltage between points A and B. (Capacitances values given are without dielectric) C = 0 pf; C = 10 pf; C = 40 pf; V = 150 V 1 B 7. Consider a sphere of nonconductor whose radius is R o and = It is positively charged with a charge density given by = A/R, where A is a constant and R the distance from the center of the sphere. Calculate the energy stored in the electric field between R = R / and R = R. o 8. Calculate the equivalent capacitance of the series combination of capacitors: C 1 = 40.0 pf, C = 60.0 pf, C = 45.0 pf. A 5.0 pf capacitor is charged to volts. The battery is then disconnected. A second capacitor, uncharged, of capacitance 60.0 pf is connected in parallel with the first. Find the new potential across the combination. A parallel plate capacitor has an area of.50 m, and a separation of mm. The dielectric material has a dielectric constant of 4.5. Calculate its capacitance in picofarads. (Be sure to use right units.) o A dielectric material has a dielectric strength of 1,500 V/mm. If it is used in a capacitor at a thickness of 0.10 mm, find the breakdown voltage for that capacitor. A potential is given by V = Ax y where A is constant and x and y are the coordinates. Find the x component of the electric field A certain material has a dielectric constant of. and a dielectric strength of volts/meter. If it is used as the dielectric material in a parallel-plate capacitor, what minimum area may the plates of the capacitor have - in order that the capacitance be F and that the capacitor be able to withstand a potential difference of 4000 volts? Given that bismuth, a semimetal, has a density of 9.80 g/cm and a carrier density of cm, calculate the number of carriers per atom. Bismuth has an atomic mass of Aluminum has an atomic weight of 7.0 and a density of.699 g/cm. Assume that aluminum has free electrons per atom. If an aluminum wire of diameter 1.00 cm carries a current of 100 amperes, find the drift velocity, and the current density. 4. Copper has a density of 8.9 g/cm and an atomic mass of 6.6. Assume that it has one free electron per atom. -6 If a copper wire of cross section 10 m carries a current of 10.0 amperes. Find the drift velocity. Find the current density.

10 4. The switch is closed for a long time and then opened. All capacitors are given without dielectric. Now a dielectric with =.5 is inserted in C. Calculate the charge and voltage on C 1 and C. With conditions as in the switch is closed. How much additional charge flows from the battery to the capacitors? = 150 V C 1 = 1.5 F C = 6.0 F 44. Given a long cylindrical nonconductor of radius R o, calculate the total energy stored in the electric field inside the cylinder per length L. The cylinder has a uniform charge density of C/m In a sample of pure silicon, the carrier density is found to be per atom. Silicon has a density of kg/m, and an atomic mass of 8.0. The silicon is in the form of a long, square rod 1.00 m on a - side and carries a current of.5 10 A along the rod. Calculate the current density. Find the drift velocity. 46. A wire has a radius is 0.5 mm and a length of 0 cm. The wire is in the center of a metal tube whose inner radius is 1.55 cm. A potential of 4000 V is applied between the wire and the tube. Calculate the energy stored in the electric field between the wire and tube. 47. In the circuit shown, initially S 1 is closed and S is open. The capacitor values are given without dielectric. Initially a dielectric with =.70 is in C. Now S 1 is opened, and then S is closed. Calculate the potential across C. With S 1 still open and S still closed, the dielectric is moved from C to C. Calculate the potential and charge on all capacitors. = 175 V C = 0 pf C 1 = 100 pf C = 150 pf 48. A cylindrical capacitor is created with the outer radius of the inner conductor R and the inner radius of the outer conductor R. A Calculate the capacitance (numerical value) of a.5 m length of this capacitor. What is the energy being stored between R = 0.50 cm and R = 1.00 cm in a.5 m length of this capacitor (numerical value). The capacitor has a potential difference of V = 175 V across it. B R = 0.5 cm A R = 1.5 cm B

11 4 49. Calculate the coefficient of the x term for the given expression: Calculate V(B) - V(A). Calculate the equivalent capacitance between a and b if all capacitors have the value C. 8 - The charge carrier density (n) for copper is e /m. The cross section of the wire is 6.0 cm. If the wire carries 11 Amps, what is the drift velocity? Calculate the current density in a copper wire with circular cross section carrying a current of 1.5 A. The diameter of the wire is.74 mm. 50. Given two conducting spheres of radius R 1 and R. A total charge Q is placed on sphere. Sphere 1 is uncharged. Now the two are connected with a wire. Calculate the charge on each sphere. Calculate the potential of each sphere. Calculate the electric field at the surface of each sphere. 51. First S 1 is closed with S open. Then S 1 is opened, and finally, S is closed. Calculate the voltage and charge for each capacitor in the final configuration = 1 Volts C 1 = 10 F C = F C = 6 F 5. Given a sphere of nonconductor with a charge density given by (for r < R o). Find Q T. Find E when r < R o. Find E when r > R o. Find the energy stored in the region o < r < R o. If < 0, what is the sign of V(0) - V(R )? Explain. o o

12 5. Calculate the capacitance of a series combination of a.75 pf and a 5.75 pf capacitor. All capacitors have the same value, C. Calculate the equivalent capacitance between points A and B. Given the function, calculate the term involving x 4 using the binomial expansion. A heavy copper wire carries a current of A. The wire is a square with sides of cm. If copper has electrons/cm that are free to move, determine the drift velocity in the wire. Find the energy stored in a parallel plate capacitor at a voltage of 165 V. The plates have an area of m and a separation of.10 mm. There is no dielectric. 54. Given the circuit shown. Calculate the equivalent capacitance between points A and B. If the battery is 175 V, what is the potential across C 1? If the battery is 175 V, what is the potential across C 5? If the battery is 175 V, what is the charge on C? C = 5.00 pf C = 6.00 pf C =.00 pf C =.00 pf C = 4.00 pf C = 7.00 pf Use at least 4 significant figures in intermediate calculations. 55. In the diagram shown the switch is closed for a long time and then opened. Now a dielectric of =.5 is inserted into C. Capacitors values are given without dielectric. Find the new voltage across C 1. Find the new voltage across C. Find the charge that flows and the direction it flows when the dielectric is inserted. = 15.0 V; C 1 = 155 pf; C = 15 pf; C = 50 pf 56. Two long conducting cylinders have the same center. The outer radius of the inner tube is.50 mm. and the inner radius of the outer tube is 11.7 mm. A potential of.75 V is applied between the inner and outer tubes. Find the energy per unit length stored between r =.50 mm and r = 4.50 mm. What is the capacitance per meter of this structure?

13 57. Calculate the equivalent capacitance between A and B if all capacitors have the same value, C. 6 Calculate the term involving x using the binomial expansion for the expression. Calculate a number or fraction for the coefficient. Calculate the current density in a copper wire with a total current of 110 Amp if the wire is circular with a diameter of.5 mm. 4 An electric potential has the form V = Ax y z, where A is a constant. Calculate the x-component of the electric field. Two capacitors of values.00 F and 4.00 F, are in series as shown. The battery is 155 V. What is the energy stored in the combination of capacitors? 58. Given the circuit shown. Calculate the equivalent capacitance between points A and B. If the battery potential is 15 volts, what is the potential across C 6? If the battery is 15 volts, what is the potential across C 4? If the battery is 15 volts, what is the charge on C? Note: Keep 4 significant figures in intermediate calculations. C =.50 pf C =.50 pf C =.00 pf C =.00 pf C = 5.00 pf C = 7.00 pf 59. Given that bismuth, a semi-metal, has a density of 9.80 g/cm and a charge 18 carrier density of /cm. Bismuth has an atomic mass of 09. Determine the number of charge carriers per atom. Calculate the drift velocity of the charge carriers in a wire of cross section 1.75 mm carrying a current of 9.50 A. 60. Given a sphere of non-conductor of radius R o with a charge density distribution given by for r < R, and = 0 everywhere else. Take as negative. o Find the total charge on the sphere, Q T. Calculate the electric field at an arbitrary point r inside the sphere. o Determine the potential difference, including its sign.

14 61. Note: If you use a sign convention different from that used in class, be sure to indicate that. For the circuit shown. How many total junctions are there? How many independent junction equations can be written? How many branches are there? How many unknown currents will be needed to analyze this circuit? Write the loop equations for loop, going clockwise around the loop. R 1 = 1000 ; 1 = 100 V; R = 900 ; = 55 V; R = For the circuit shown, use the values given below. Calculate the current in R, including sign. The arrows show the positive direction. Find the magnitude of the potential across R 1. Find the current, magnitude and sign, through R 4. The arrow shows the positive direction. R = 400 R = 00 R = 150 R = R = 800 = 115 V = 17.0 V = 95.0 V 6. Use the numerical values below for the components in the circuit. At t = 0 the switch is closed after being opened for a long time. Calculate the current in R when 0.75 time constants have elapsed after t = 0. Calculate the current in R 1 at t =. Calculate the charge on the capacitor after 0.75 time constants have elapsed since t = 0. Using the full loops and junctions, and NO shortcuts, calculate the time constant for charging the capacitor. (Use the currents indicated.) -6 = 15 V; R 1 = 000 ; R = 500 ; R = 1750 ; C = F 64. Calculate the capacitance of the series combination of a 4.0 pf capacitor and a.0 pf capacitor. Calculate the capacitance of the parallel combination of a.0 pf capacitor and a 1.5 pf capacitor. An electron is subjected to an electric field of 70 V/m. Find its acceleration. A 1.5 volt battery is connected across a 100 pf capacitor. Find the charge on the capacitor. The electric potential in a region of space is given by V = Cx. Find the electric field. 65. Calculate the capacitance of two concentric metal spheres, one of radius a and the other of radius b. 66 When the switch is closed, C 1 and C have no dielectric and C has a dielectric totally filling the capacitor with K = The values of C 1 and C given are without dielectric, C is with the dielectric. Find the potential and charge on each capacitor. Now the switch is opened and the dielectric is moved from C to C 1. Find the new values of the charge and potential for each capacitor. V = 65.0 V C = pf C 1 =.00 pf C = 8.00 pf

15 67. In the arrangement shown, S A is originally closed and S B is open. Then S A is opened followed by closing S. B With S B closed, find the charge on each capacitor. With S B still closed, find the voltage across each capacitor. Leaving S B closed, close S A. How much additional charge flows through S? A V = 150 V; C 1 = 5 pf; C = 0 pf; C = 50 pf 68. In the diagram shown the switch is closed for a long time and then opened. Now a dielectric of = 4.00 is inserted into C. Find the new voltage across C 1. Find the new voltage across C. Find the charge that flows and the direction it flows when the dielectric is inserted. = 1.0 V; C 1 = 155 pf; C = 15 pf; C = 50 pf 69. In the diagram shown = 150 V 1, C 1 = 0 pf and C = 40 pf. The switch S is closed and then a dielectric of dielectric constant = 4.00 is inserted in C. Find the potentials and charges on the capacitor. Now the switch is opened and the dielectric is moved from C to C 1. Find the new potentials and charges in the capacitors. If you cannot do part use V 1, V, Q 1 and Q as starting charges and potentials for part. 70. In the arrangement shown, both switches are initially open and all capacitors are fully discharged. Now close S 1 while leaving S open. Then open S 1 with S still open. Finally, close S with S 1 remaining open. Work parts and with the switches in this state. Calculate the charge on each capacitor. Calculate the voltage across each capacitor. Leaving S closed with the charges and voltages as in and, close S 1. How much additional charge flows from the battery to the capacitors? V = 1.5 V; C 1 =.00 pf; C = 4.00 pf; C =.00 pf 71. In the arrangement shown both switches are initially open and all capacitors are fully discharged. First, close S 1 with S open. Then open S 1, leaving S open. Finally, close S with S 1 still open. Work part with the switches in this state. Capacitance values are without dielectric. Calculate the charge on the potential difference across each capacitor. With S closed and S 1 open, as in, insert a dielectric with = 5.00 into C. Recalculate the charge and potential difference for each capacitor. V = 110 V; C 1 = 7.5 pf; C =.00 pf; C = 5.00 pf 7. A conducting sphere has a charge Q = C and a radius R o, where R o = 0.7 m. Calculate the energy stored in the electric field between R = R and R = R. -6 o o

16 7. Initially all capacitors are uncharged. Then switch S 1 is closed leaving S open. Calculate the potential and charge for capacitors C 1 and C. Now S 1 is opened, and then S is closed. Calculate the potential and charge on each capacitor. With S still closed, close S 1 and calculate the additional charge that flows from the battery to the capacitors. C = 10 F C = 0 F 1 B C = 70 F V = 15 V 74. A wire carries a current of 500 A. The cross section of the wire is a square, with each edge of the 8 square having a length of cm. If copper has electrons/m free to move, calculate the drift velocity of the free electrons in the wire. Calculate the effective resistance between points a and b. All resistors are A resistor carrying 1.07 A of current is dissipating 47. watts. Calculate its resistance. -8 Copper has a resistivity at room temperature of ohm m. Find the resistance at room temperature of.00 miles of copper wire whose diameter is 1.5 mm. 75. If each resistor is.00 ohms, calculate the effective resistance between and. A heavy copper wire carries a current of 1500 Amps. The wire is square with each side cm. If 8 copper has electrons/m free to move, calculate the drift velocity in this wire. A 1,500 ohm resistor is dissipating 0.75 watts of power. Find the current in this resistor. 76. Calculate the effective capacitance between a and b. All capacitors have a value of 50 pf. Silicon has an atomic mass of 8.0 and a density of kg/m. If a sample of silicon has a charge carrier density of cm, how many carriers per atom does it have? 77. A current of 150 A is flowing in an aluminum wire of diameter.00 mm. Aluminum has three free electrons per atom, an atomic mass of 7.0 and a density of.7 10 kg/m. Find the drift velocity. Find the effective resistance between a and b. All resistors have the same value.

17 Calculate the effective capacitance between a and b if all capacitors have the value of 175 Pf Given a simple circuit consisting of a resistance of ohms and a capacitance of 6.0 pf, find the time constant. Given an RC circuit. The capacitance begins to discharge at t = 0. If the charge on the capacitor is C at t = 0, what is it at t = RC? A current of 1.5 amperes passes through a resistance of 100 ohms. What is the power dissipated? 17 Bismuth has an atomic mass of If it has.0 10 charge carriers per cm and a density of 9.80 g/cm, how many charge carriers per atom does it have? Calculate the effective resistance between points a and b Given a simple circuit consisting of a resistance of.7 10 ohms and a capacitance of 15,000 pf, find the time constant. Given an RC circuit. The capacitor is discharged at t = 0. The capacitor is charged through the resistance R from a battery of 100 V. If the capacitance is 000 pf, what is the charge on the capacitor at t = 1.5 RC? A current of 15 amperes passes through a resistance. The potential across the resistance is 110 volts. What is the power dissipated in the resistance? 18 Antimony has an atomic mass of 11.8 and a density of 6.68 g/cm. If antimony has.0 10 charge carries per cm, how many carriers per atom does it have? Calculate the effective resistance between points a and b. 80. A parallel plate capacitor is constructed with a plate area of 0.40 m and a plate separation of mm. A dielectric plate, of dielectric constant.5, completely fills the volume between the plates. The capacitor is connected to a battery of 90 V, and then disconnected. Then the dielectric plate is removed. What is the charge on the plates? What is the potential between the plates. What is the electric field between the plates? If the capacitor is now reconnected to the battery, how much charge flows, and in which direction? 81. Given the arrangement of capacitors shown with the values shown. Calculate the effective capacitance between A and B. If a potential of 150 volts is applied between A and B, find the charge on C 4. If a potential of 150 volts is applied between A and B, find the potential across C. C 1 = F; C = 5.00 F; C = F C = 1.00 F; C = F; C = 0.00 F 4 5 6

18 8. Calculate the equivalent resistance between a and b for the network given. All resistors have the same value, R. -1 A resistor of R = 150 is dissipating.7 10 W. Calculate the current in it. -8 Aluminum has a resistivity at room temperature of.8 10 ohm m. Calculate the cross sectional area for an aluminum wire if it is to have a resistance of 1.00 per mile. For the network shown, calculate the potential across R. R 1 = 15 R = 150 R = 75 R 4 = 45 = 100 V 4 8. In the circuit shown the switch is open for a long time and then closed at t = 0. At t = 0, calculate the voltage across R. Calcualte the voltage across R at t =. After 1.00 time constant has elapsed after t = 0, calculate the charge on the capacitor. Calculate a numerical value for the time constant. Show all work with no short cuts from more advanced classes. R = 150 R = 115 C = 10 F 1 R = 175 = 100 V 84. A wire of radius r = mm is at the center of a conducting cylinder of inner radius R = 1.5 cm. There is a potential difference of 55 V between the wire and the cylinder. Calculate the energy stored in a.00 m length of this structure between r and R/. 85. Calculate the equivalent capacitance between and. All capacitors are.00 pf. Calculate the effective resistance between and. All resistors have a value of 100 ohms. Calculate the drift velocity for a current of 1.0 A in a circular cross section copper wire of radius mm. Take the charge carrier density as m.

19 Calculate the power being dissipated in the 150 ohm resistor. 18 Antimony has an atomic mass of 11.8 and a density of 6.68 g/cm. If antimony has charge carriers per cm, how many carriers per atom does it have? 86. A copper rod, with circular cross section, has a radius of 1.00 cm. If it carries a current of 47,500 A, calculate the current density. A 75 resistor dissipates power at the rate of 4.40 Watts. Calculate the current. Calculate the equivalent capacitance between points A and B. All C's are 5 pf. Calculate the equivalent resistance between X and Y. Calculate the drift velocity for a current of 1,000 A in a copper rod with square cross section 1.00 cm 9 - on a side. Take the charge carrier density as m. 87. For the circuit shown, calculate the time constant. R = C 1 = 5.0 pf C = 40.0 pf Calculate the capacitance of a parallel plate capacitor if the plates are square, with sides of 0.50 m, a separation of 0.0 mm, and a dielectric of = 4.50 filling the space between the plates. A copper wire has a diameter of.0 mm. If it carries a current of 17. A, calculate the magnitude of the current density in A/m. Antimony (a semimetal) has atomic mass of 11.8 and a density of 6.68 g/cm. If we assume it has charge carriers per atom, what is the density of charge carries per m? All resistances in the network have a resistance R. Calculate the effective resistance between a and b.

20 88. For the circuit shown all capacitance values are given WITHOUT DIELECTRIC. Initially, all capacitors are discharged. A dielectric with = 4.00 is inserted in C, and then the switch is closed. After a time long enough to equilibrate, the switch is opened. Now the dielectric is moved from C to C. The switch remains open. Calculate the charge and potential on each capacitor before the dielectric is moved from C to C. Calculate the charge and potential on each capacitor after the dielectric is moved from C to C. = 150 V C = 15.0 F C 1 =.0 F C = 10.0 F 89. For the circuit shown, and the values given, calculate: the potential across R 1; the potential across R ; the power being dissipated in R ; the power being dissipated in R. = 150 V R = 650 R = 10 R = 70 R = = 100 V R = 50 R = 50 R = 80 R = If all capacitors have the same value, C, calculate the effective capacitance between a and b. Silicon has an atomic mass of 8.0 and a density of. g/cm. If impurities are added so there is charge carriers per atom, calculate the number of charge carriers per cm. A copper wire carries a current of 157 A. If the diameter of the wire is.00 mm, what is the current density? -4 If the current in is carried by electrons and the drift velocity is m/s, what the density of charge carriers? If 60 volts is applied between A and B, calculate the charge on the 15 pf capacitor. 91. A sphere of non-conductor has a volume charge density give by = A/R, for R < R o, where R o is the radius of the sphere. Calculate the electric field at a point R = R A, where R A < R o (inside the sphere). Calculate the magnitude of the electric potential difference between the surface of the sphere and an arbitrary point in the interior that is a distance R B from the center. Evaluate the constant A in terms of the total charge on the sphere.

21 9. Calculate the effective resistance between A and B if all resistors have the value of.00 ohms. Calculate the effective value of the capacitance between A and B if all capacitors have the value 0.0 pf. If = 75.0 V, what is the power being dissipated in the 5 resistor? Calculate the drift velocity (in m/s) in a silver wire with current = A that has a circular cross section with r = m. - Silver has electrons/cm that are mobile. A parallel plate capacitor is constricted using a material whose dielectric constant is 4.7. If it is necessary to have the plate separator at 1.7 mm, what is the area needed to give C = 17.0 pf? 9. For the network shown, S is closed for a long time with S open. S is then opened and S closed. S remains open With S closed, find the potential on C 1. With S closed, calculate the charge on C 4. With S closed, what is the charge on C? V batt = 10 V C 1 = 50 pf C = 150 pf C = 100 pf C = 5 pf Given the network shown. How many total junctions are there? How many unknown currents are there? Write down the loop equations for interior loops A and B. Traverse the loops clockwise. Write down the junction equation for junction C. How many loop equations will be needed for a complete solution?

22 95. Calculate the equivalent capacitance between a and b. All capacitors are 7.00 F. Find the effective resistance between a and b if all resistors are Determine the power being dissipated in the 150 resistor. V = 15 V. -8 A given metal has a resistivity of ohm m. If a wire is to have a resistance of.00 per mile, calculate its cross-sectional area. The 150 pf capacitor is charged to 100 volts. The battery is removed and then the switch is closed. Calculate the charge on the 100 pf capacitor. 96. In the network shown, all resistors have a resistance of 5.00 ohm. Calculate the effective resistance between a and b. If a 75.0 volt potential is applied between a and b, what is the current in the resistor labeled x? With the same 75.0 volt potential between a and b, calculate the potential across y.

23 97. Given the network shown. Use the sign convention used in class, or make clear to the grader what convention you use. How many total junctions are there? How many total unknown currents are there? Write the junction equation for junction C using the current designations given. Write the loop equation for interior loop (A) going clockwise around the loop. Use the current designations given. 98. For the circuit shown, all capacitance values are given WITHOUT DIELECTRIC. Initially there is a dielectric with =.70 in C 4 only. The switch S is closed, and then opened. With the switch open the dielectric is removed from C. After the dielectric is removed from C and with the switch still open, find: 4 4 the charge and potential on C 1; the charge and potential on C ; the charge and potential on C. C 1 = 450 F; C = 0 F; C = 10 F; C = 85 F; V = 15 V 4 B Calculate the equivalent capacitance between a and b if all capacitors have the same value, C. Calculate the equivalent resistance between a and b if all resistors have the same value, R. A capacitor of value 16.5 pf is charged to a potential of 50 V. Calculate the energy stored in this capacitor. Assume a value for the electric field near the earth's surface of 150 V/m, directed vertically. Calculate the energy stored in a volume of.50 m near the earth's surface. -8 The resistivity of copper is ohm m. Calculate the resistance of 1500 m of round copper wire of diameter 1.00 mm.

24 100. For the circuit shown: How many TOTAL junctions are there? How many mathematically independent junction equations can be written? Write the junction equations for junctions A and B in the drawing. How many mathematically independent loop equations can be written? Write loop equations for loops I and II in the drawing. For ease in grading, go clockwise around the loop. [Tell us if you use a sign convention different from that used in class.] 101. The capacitor values above are given without dielectric. The switch is closed and a dielectric with =.00 is inserted in C. With the switch still closed, calculate the potential and charge on C. Now the switch is opened and remains open. The dielectric is moved from C to C 1. Find the new value of the potential and charge on C. Calculate the potential between a and b with the switch in the same position as in. = 10 V C = 0.0 F C 1 = 1.0 F C = 6.00 F 10. A copper wire has a diameter of mm. It carries a current of 4.75 A. Calculate the current density. Silicon has a density of kg/m and an atomic mass of 8.0. If, in a given sample of silicon 5 there are.5 10 carries per atom, calculate the density of charge carriers per m, that is, n. Calculate the equivalent resistance between a and b if all resistors have the same value R. Calculate the charging time constant for the circuit shown. R = 7500 ; C = 5.0 F Given an RC circuit. The charge on the capacitor is C at t = 0. It begins to discharge at 1 t = 0. What is the charge at t = 1.75?

25 10. Given the network shown. Use the sign convention used in class, OR make it clear to the grader what sign convention you use. How many TOTAL junctions are there? How many total unknown currents are there? How many loop equations will be needed? Write the junction equation for junction (A) using the current designations given. Write the loop equation for the interior loop designated (B). Start at the lower left corner and go clockwise around the loop For the circuit shown, the switch is closed at t = 0, after being open for a long time. Calculate the charge on the capacitor at t =. Calculate the current in R 1 at t =. Calculate the time constant (numerical value) for charging the capacitor. After the switch has been closed for a long time, it opened at a new t = 0. Write an expression for the voltage on the capacitor as a function of time with all quantities evaluated numerically. = 175 V; R 1 = 50 ; R = 475 ; C = 65.0 pf 105. Initially the switch is closed. Calculate the potential and charge on each capacitor Now the switch is opened and dielectric with = 4.00 is inserted in C. None of the other capacitors have any dielectric. Calculate the potential and charge on all of the capacitors in this new situation. = 150 V; C 1 = 100 pf; C = 5.0 pf; C = 5.0 pf; C 4 = 65.0 pf 106. Silver has a density of 10.5 g/cm and an atomic mass of 108. Assume that silver has one free charge carrier per atom. If a silver wire of diameter.00 mm carries a current of 10.7 A, find the current density; the number of charge carriers per cubic meter for silver; the drift velocity Given the network shown. Use the sign convention used in class, OR make it clear to the grader what sign convention you use. How many TOTAL junctions (nodes) are there? How many total unknown currents are there? How many mathematically independent loop equations will be needed? Write the junction equation for junction (A) using the current designations given. Write the loop equation for the interior loop designated (B). Start at the lower left corner and go clockwise around the loop.

26 108. For the circuit shown and using the methods discussed in class: Calculate the current in R 1. Calculate the potential across R. Calculate the current in R. R = 500 R = 700 R = 00 R 4 = 100 R = = 100 V = 50 V = 150 V 109. Each resistor has a value of ohms. Calculate the equivalent resistance between a and b. Calculate the equivalent capacitance of three capacitors.0, 5.0 and 9.0 F in series. A copper wire has a diameter of 1.75 mm. It carries a current of 4.5 A. Calculate the current density. Calculate the charging time constant if R = and C has the value of 1.8 pf. A capacitor of 4.75 F is charged to 150 V. The battery is disconnected and then a dielectric is inserted. The dielectric is = 4.1. Calculate the potential in the capacitor after the dielectric has been inserted Only currents needed for and are shown. For the network shown, how many mathematically independent junction (node) equations can be written? For the network shown, how many mathematically independent loop equations can be written? Write the loop equation for the loop A. Use the sign conventions discussed in class, or make a clear statement that you are using a different convention. Write the junction equation at the junction labeled B.

27 111. For the circuit shown, the capacitor is initially uncharged. The switch is closed at t = 0. Calculate the current in R 4 at t = 0. Calculate the current in R at t =. Calculate the charge on the capacitor at t =. Calculate the charge on the capacitor afer 1.5 time constants have elapsed from t = 0. Using the methods discussed in class, calculate the time constant for charging this capacitor. 11. For the circuit given, circle and number number all of the junctions. How many independent junction equations can be written? How many loop equations are needed in addition to the junction equations to completely solve for the unknown currents? Write down the junction equation for the junction labeled A. Use the current labels and directions given. Write down the loop equations for the interior loops labeled (1) and (). Start each at A and go clockwise. Signs are important. Use the current labels and directions given. 11. How many independent junction equations can be written for the circuit shown? How many independent loop equations? Using the current labels and direction given, write a set of junction and loop equations that can be used to analyze the circuit. (To simplify grading, go clockwise around your loops.) 114. In the circuit shown, the capacitor is initially uncharged, and the switch S is closed at time t = 0. Calculate the charge on the capacitor at t =. Find the voltage across the capacitor when t = Calculate the current in R 1 at t = 0. Find the time constant of the circuit (numerical value). Note: No short cuts learned in other courses are permitted. R 1 = 150 k ; R = 175 k ; R = 15 k ; C = 50 pf; = 55.0 V 115. In the circuit shown, the switch is closed at t = 0. Calculate the numerical values for the following quantities. (No short cuts such as Thevenin's Theorem, or other material from advanced courses permitted.) The voltage on the capacitor at t =. The voltage on the capacitor at t =. The voltage across R 1 at t = 0. The time constant for the circuit. = +150 V; R 1 = ; R = ; R = 50.0 ; R = 50.0 ; C =.00 mf 4

28 116. In the circuit shown, switch S is connected to point A for a long time. What is the potential at C? Now S is switched to point B. What is the potential at C after a long time? How much charge flows through R from the time the switch is thrown to B until current stops flowing in R? Find the current in R as a function of time. Be sure to put numerical values for all quantities, including the time constant. 5 C 1 = 10 pf; C = 0 pf; C = 10 pf; C 4 = 0 pf; R = 10 ; = 1.0 V 117. The capacitor is originally uncharged. The switch is placed at A for s. Then it is moved to B. Call this t = 0. Calculate the charge on the capacitor at t = s. Calculate the current through R at t = s. At t = s, the switch is moved back to A. How much charge is added to C before equilibrium is attained? = 1.0 V; C = 6.75 ; R 1 = 1,500 ; R = 17,00 R =, In the circuit shown the capacitor is initially discharged. The switch is placed in position A for 5.0 seconds, and then moved to position B. Calculate the potential across R 7.0 seconds after the switch is moved to B. Calculate the charge on the capacitor 7.0 seconds after the switch is moved to B. The capacitor is charged for 5.0 seconds with the switch at A. Then the switch is opened and a dielectric of =.00 is inserted in the capacitor. Then the switch is moved to B. Now calculate the charge and the potential on the capacitor 7.0 seconds after the switch is thrown to B = 1.0 V; C = F; R 1 = ; R = ; R = In the circuit shown the switch S is closed at t = 0. Calculate the current in R just after the switch is closed. Calculate the current in R a very long time after the switch is closed. Calculate the voltage across the capacitor a very long time after S is closed. Calculate the charge on the capacitor 15 s after the switch is closed. No short cuts learned in other classed are allowed. Part requires the full solution as discussed in class. -6 R 1 = 150 ; R = 50 ; R = 50 ; = 15 V; C = F

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