1. Two point charges. Two identical point charges 1 m apart from
each other in free space are experiencing a repulsion force of 1 N each.
What is the magnitude of each charge? Get solution
2. Two point charges. Two small identical spheres have charges of +20 nC and −5 nC, respectively. (a) What is the force between them if they are apart by 10 cm? (b) The two spheres are brought into contact and then separated again by 10 cm. What is the force between them now? Get solution
3. Two suspended charges. Two small, identical, electrically charged conducting spheres of mass 2.5 g and charge +150 nC each are suspended by weightless strings of length 12 cm each, as shown in Figure. Calculate the deflection angle θ.Figure Two suspended charges. Problem.... Get solution
4. Zero force. (a) Three point charges of Q1 = +40 nC, Q2 = −20 nC, and Q3 = +10 nC are all situated on the x axis in such a way that the net force on charge Q2 due to Q1 and Q3 is equal to zero. If Q1 and Q2 are located at points (−2, 0, 0) and (0, 0, 0), respectively, what is the location of charge Q3? (b) Q3 is moved to a different position on the x axis such that the force on itself due to Q1 and Q2 is equal to zero. What is the new position of Q3? Get solution
5. Three charges. Three identical charges of charge Q are located at the vertices of an equilateral triangle of side length a. Determine the force on one of the charges due to the other two. Get solution
6. Three point charges. Three point charges of values +150 nC, +100 nC, and +200 nC are located at points (0, 0, 0), (1, 0, 0), and (0, 1, 0), respectively. (a) Find the force on each charge due to the other charges. Which charge experiences the largest force? (b) Repeat part (a) for −100 nC as the second charge. Get solution
7. Two point charges. Two point charges of +Q and −Q are located at (0, 0, 0) and at (0, 4a, 0) respectively. Find the electric field at P1(0, 0, 3a) and at P2(0, 4a, 3a). Sketch the orientations of the fields. Get solution
8. Zero field from three charges. Two point charges of +Q each are located at (10, 0) cm and (−10, 0) cm, while a third one of charge −2Q is at (0,−10) cm. Find the coordinates of the point where the electric field is zero. Get solution
9. Three point charges. Two point charges of +10 nC each are located at points (0.46, 0, 0) and (−0.46, 0, 0), respectively. (a) Where should a third point charge of +15 nC be placed such that E = 0 at point (0, 1, 0)? (b) Repeat part (a) for a third charge of −15 nC. (c) With the −15 nC charge located as you determined in part (b), is there another point at which the electric field E = 0? If so, specify this point. Get solution
10. Four point charges on a square. Four point charges of +50 nC each are located at the corners of a square of side length 10 cm located on the xy plane and centered at the origin. (a) Find and sketch the electric potential on the z axis. (b) Find and sketch the electric field on the z axis. Get solution
11. Six point charges on a hexagon. Six identical point charges of +25 nC each are situated in space at the corners of a regular hexagon whose sides are each of length 6 cm. (a) Find the electric potential at the center of the hexagon. (b) Determine the energy required to move a point charge of −25 nC from infinity to the center of the hexagon. Get solution
12. Two straight-line charges. Consider two uniformly charged wires, each of length 1 m and a total charge +100 nC, with their ends separated by 1 m, as shown in Figure. (a) Find the electric potential ϕ at the point P, midway between the two wires. (b) Find the electric field E at point P.Figure Two straight-line charges. Problem.... Get solution
13. Seven point charges on a cube. Seven identical point charges of +10 nC each occupy seven of the eight corners of a cube 3 cm on each side. Find the electric potential at the unoccupied corner. Get solution
14. Two line charges. A uniform line charge of ρl1 = −4π × 8.85 pC-m−1 is located between the points (−5, 0) and (−2, 0) m and another such line of positive charge (i.e., ρl2 = 4π × 8.85 pC-m−1) between (5, 0) and (2, 0) m, as shown in Figure. (a) Find the electric potential ϕ at (1, 0) m. (b) Find the electric field E at the same point. At which points, if any, is the electric field E zero? At which points, if any, is the electric potential ϕ zero?Figure Two line charges. Problem.... Get solution
15. Circular ring of charge. A total charge of Q1 is distributed uniformly along a half-circular ring as shown in Figure. Two point charges, each of magnitude Q2, are situated as shown. The surrounding medium is free space. (a) Find Q2 in terms of Q1 so that the potential ϕ at the center of the ring is zero. (b) Find Q2 in terms of Q1 so that the electric field E at the center of the ring is zero.Figure Circular ring of charge. Problem.... Get solution
16. Semicircular line charge. A thin line charge of density ρl is in the form of a semicircle of radius a lying on the xy plane with its center located at the origin, as shown in Figure. Find the electric field at the origin for the cases in which (a) the line charge density ρl = ρ0 is a constant and (b) the line charge density varies along the semicircular ring as ρl = ρ0 sin φ.Figure Semicircular line charge. Problem.... Get solution
17. Charge on a hemisphere. The curved surface of a hemisphere of radius a centered at the origin carries a total charge of Q uniformly distributed over its curved surface, as shown in Figure. (a) Find the electric potential on the z axis. (b) Find the electric field on the z axis. (c) Repeat parts (a) and (b) if the charge Q is uniformly distributed throughout the volume of the hemisphere.Figure Charge on a hemisphere. Problem.... Get solution
18. Sheet of charge with hole. An infinite sheet of uniform charge density ρs is situated coincident with the xy plane at z = 0. The sheet has a hole of radius a centered at the origin. Find (a) the electric potential ϕ and (b) the electric field E at points along the z axis. Get solution
19. Spherical charge distribution. A charge density of...where K and b are constants, exists in a spherical region of space defined by 0r a. (a) Find the total charge in the spherical region. (b) Find the electric field at all points in space. (c) Find the electric potential at all points in space. (d) Show that the potential found in part (c) satisfies the equation ∇2 = ϕ−ρ(r)/∊ 0 for both r a and r >a. Get solution
20. The electron charge density in a hydrogen atom. According to quantum mechanics, the electron charge of a hydrogen atom in its ground state is distributed like a cloud surrounding its nucleus, extending in all directions with steadily decreasing density such that the total charge in this cloud is equal to qe (i.e., the charge of an electron). This electron charge distribution is given by...where a is the Bohr radius, a ≃ 0.529 × 10−10 m. (a) Find the electric potential and the electric field due to the electron cloud only. (b) Find the total electric potential and the electric field in the atom, assuming that the nucleus (proton) is localized at the origin. Get solution
21. Spherical shell of charge. The space between two concentric spheres of radii a and b (a b) in free space is charged to a volume charge density given by...where K is a constant. (a) Find the total charge in the shell. (b) Find the electric field at all points in space. (c) Find the electric potential at all points in space. (d) What happens if b → a? Get solution
22. Space charge between parallel plates. The space between two perfectly conducting parallel plates is filled with space charge (i.e., free charge) of density given as:...where d is the separation distance of the plates. Other than the space charge, the medium between the plates is air, with permittivity ∊0. The upper and lower plates, located at z = 0 and z = d are kept at potentials of ϕ = 0 and = V0, respectively. Determine expressions for the potential (z ) and the electric field E(z ) between the plates. Get solution
23. Spherical charge distribution. A spherical charge distribution exists in free space in the region 0r a given by...(a) Find the total charge. (b) Determine E everywhere. (c) Determine ϕ everywhere. (d) Sketch both |E| and ϕ as a function of r. Get solution
24. Spherical charge with a cavity. A spherical region of radius b in free space is uniformly charged with a charge density of ρ = K, where K is a constant. The sphere contains an uncharged spherical cavity of radius a. The centers of the two spheres are separated by a distance d such that d + a b. Find the electric field inside the cavity. Get solution
25. Charge on a hollow metal sphere. A hollow metal sphere of 20 cm diameter is given a total charge of 1 μC. Find the electric field and the electric potential at the center of the sphere. Get solution
26. Electric field. An electric field in empty space is given as:...Determine the energy required to move a positive unit test charge in the presence of this electric field from the point (0, 0) to (1, 1) along the shortest straight line in the x-y plane.Determine the energy required to move the same test charge from point (1, 1) back to (0, 0) but via a different path, by first going fro (1, 1) to (1, 0) and then from (1, 0) to (0, 0). Get solution
27. A 1-farad capacitor. To get an idea about the physical size of a 1-F capacitor, consider a parallel-plate capacitor with the two metal plates separated by 1 mm thickness of air. Calculate the area of the metal plates needed so that the capacitance is 1 F. Get solution
28. Gate oxide capacitance of a MOS transistor. A basic MOS transistor consists of a gate conductor and a semiconductor (which is the other conductor), separated by a gate dielectric.Consider a MOS transistor using silicon dioxide (SiO2) (∊r = 3.9) as the gate oxide. The gate oxide capacitance can be approximated as a parallel-plate capacitor. The gate oxide capacitance per unit area is given by...where ∊ox and tox are the permittivity and the thickness of the gate dielectric. (a) If the thickness of the SiO2 layer is 2 × 10−6 cm, find the gate oxide capacitance per unit area.(b) If the length and the width of the gate region are L = 5 × 10−4 cm and W = 2 × 10−3 cm, respectively, find the total gate capacitance. Get solution
29. RG 6 coaxial cable. A coaxial cable (RG 6) designed for interior use, such as connecting a TV set to a VCR, has a per-unit-length capacitance listed as 17.5 pF/ft. If the relative dielectric constant of the insulator material in the cable is ∊r ≃ 1.64, find the ratio of the inner and outer radii of the insulator. Get solution
30. Radius of a high-voltage conductor sphere. Consider an isolated charged metallic conductor sphere in a dielectric medium at an electric potential of 500 kV. Calculate the minimum radius of the sphere such that dielectric breakdown will not occur if the surrounding dielectric is (a) air (EBR =3 MV-m−1), (b) a gaseous dielectric such as sulphur hexafluoride (SF6) (∊r ≃ 1 and EBR = 7.5 MV-m−1), (c) a liquid dielectric such as oil (∊r = 2.3 and EBR = 15 MV-m−1), and (d) a solid dielectric such as mica (∊r = 5.4 and EBR = 200 MV-m−1). Get solution
31. Parallel-plate capacitor. A parallel-plate capacitor is constructed from two aluminum foils of 1 cm2 area each placed on both sides of rubber (∊r = 2.5 and EBR = 25 MV-m−1) of thickness 2.5 mm. Find the voltage rating of the capacitor using a safety factor of 10. Get solution
32. Energy in a capacitor. A 9-V battery is connected across a parallel-plate air-filled capacitor. The battery is subsequently removed, and a block of solid dielectric (∊ = 2∊0) is inserted between the plates. (a) What is the voltage across the capacitor after the introduction of the dielectric? (b) Compare the total electrostatic energy stored in this capacitor before and after the introduction of the dielectric. Comment and explain any differences. Neglect all fringing effects. Get solution
33. Coaxial capacitor. Consider a coaxial capacitor as shown in Figure. Given a = 5 mm, l = 3 cm, and the voltage rating of the capacitor to be 2 kV with a safety factor of 10, what is the maximum capacitance that can be designed using (a) oil (∊r = 2.3 and EBR = 15 MVm −1) and (b) mica (∊r = 5.4 and EBR = 200 MV-m−1).Figure Coaxial capacitor. Problem.... Get solution
34. Coaxial capacitor with two dielectrics. A coaxial capacitor consists of two conducting coaxial surfaces of radii a and b (a b). The space between is filled with two different dielectric materials with relative dielectric constants ∊1r and ∊2r , as shown in Figure. (a) Find the capacitance of this configuration. (b) Assuming that l = 5 cm, b = 3a = 1.5 cm, and oil and mica are used, calculate the capacitance. (c) Redo part (b) assuming that only oil is used throughout.Figure Coaxial capacitor with two dielectrics. Problem.... Get solution
35. Capacitor with spacers. The cross-sectional view of an air-filled coaxial capacitor with spacers made out of material with permittivity ∊ is shown in Figure. (a) Find the capacitance of this coaxial line in terms of∊, a, b, and φ. (b) If the spacers are to be made out of mica (∊ = 6∊0), determine the angle φ such that only 10% of the total energy stored by the capacitor is stored in the spacers. (c) Consider the capacitor without the spacers (i.e., φ = 0). For a given potential difference V0 between the inner and outer conductors and for a given fixed value of b, determine the inner radius a for which the largest value of the electric field is a minimum.Figure Coaxial capacitor with spacers. Problem.... Get solution
36. Earth capacitor. Consider the earth to be a large conducting sphere. (a) Find its capacitance (the earth’s radius is ∼6.371 × 106 m). (b) Find the total charge and energy stored on the earth (take the electric field on the surface of the earth to be 100 V-m−1). (c) Find the maximum charge and energy that can be stored on the earth. Get solution
37. Coaxial capacitor with variable ∊. A coaxial capacitor of inner radius a and outer radius b is filled with a dielectric material whose relative permittivity varies as ∊r = 10 a/r over the region from r = a to r = b. Find the capacitance per unit length and compare with the capacitance of the same coaxial cable when filled with air. Get solution
38. Planar charge. A surface charge distribution ρs (x, z ) exists on the x-z plane, with no charge anywhere else (i.e., ρ = 0 for |y| >0). Which of the following potential functions are valid solutions for the electrostatic potential in the half-space y >0, and what is the corresponding charge distribution ρs (x, z ) on the x-z plane?... Get solution
39. Parallel power lines. An isolated pair of parallel power lines a distance of d1 apart have a potential difference of VAB and are located a distance h above a pair of telephone wires, as shown in Figure. The parameter values are d1 = 1 m, a = 2 cm, VAB = 440 V, h = 60 cm, and d2 = 15 cm. (a) Find the direction and magnitude of the electric field at points 1 and 2. Take the midpoint between the power lines as the origin of your coordinate system. (b) Determine the potential difference ϕ12 between points 1 and 2.Figure Parallel power lines above telephone lines. Problem.... Get solution
40. Field under high-voltage line. Many 60 Hz high-voltage transmission lines operate at an rms alternating voltage of 765 kV. (a) What is the peak electric field at ground level under such a line if the wire is 12 m above the ground? (b) What is the peak potential difference between the head and feet of a 6-ft tall person? (c) Is the field sufficient to ignite a standard (110 V) fluorescent lamp of 2 ft length? Get solution
41. Capacitance of the two-wire line with dielectric sleeve. A two-wire line consists of two metallic conductors of radius a enclosed by dielectric (permittivity _) sleeves of radii b separated by a distance d as shown in Figure, with d a. The space surrounding the dielectric sleeves is air, with permittivity _0. (a) Determine the capacitance per unit length of this two-wire line. (b) The values of wire radius and separation distance are given to be a = 1 cm and d = 20 cm, respectively, while the radius of the dielectric sleeve is b = 5 cm.If the dielectric sleeve is made of mica (∊ = 5.4∊0, EBRmica = 200 MV-m–1), determine the maximum operating voltage of the two-wire line capacitor. Note that the breakdown field for air is EairBR = 3 MV-m−1.Figure Two-wire transmission line. Problem.... Get solution
42. Thundercloud fields. A typical thundercloud can be modeled as a capacitor with horizontal plates with 10 km2 area separated by a vertical distance of 5 km. Just before a large lightning discharge, the upper plate may have a total positive charge of up to 300 C, with the lower plate having an equal amount of negative charge. (a) Find the electrostatic energy stored in the cloud just before a discharge. (b) What is the potential difference between the top and bottom plates? (c) What is the average electric field within the cloud? How does this value compare to the dielectric breakdown field of dry air (3 MV-m−1)? Get solution
43. Two conducting spheres. Consider a pair of small conducting spheres with radii a, b, small compared with the separation distance d between their centers (i.e., a, b ≪ d). (a) Determine the electrostatic energy stored by this configuration, assuming that the spheres with radii a and b carry charges of Q and −Q, respectively. Your answer should depend on d. State all assumptions. (b) Repeat part (a) assuming that the spheres with radii a and b carry charges of +Q and +2Q, respectively. Get solution
44. Voltage-controlled actuator. Consider a voltage-controlled parallel-plate actuator, as shown in Figure, with plate area A = 500 × 500μm2, equilibrium separation distance d0 = 2 μm, and effective spring constant k = 0.01 N-m−1. The capacitor is filled with free space. (a) What is the usable voltage range of this actuator? (b) What is the spacing d between the top and bottom capacitor plates when a voltage of 0.05 V is applied? Get solution
45. Air-bag accelerometer. The comb-drive accelerometer depicted in Figure is to be used in an integrated circuit that is responsible for deploying an air-bag in the event of an automobile crash. Assuming a sufficient overlap between the plates such that the effects of fringing fields remain unchanged and effective spring constant k = 0.01 N-m−1, determine the change in capacitance during a deceleration of 40g, where g ≈ 9.8 m-s−2. Assume the capacitor is filled with free space, there are N = 100 capacitor plate pairs, and the following dimensional parameters: L = 200 μm, w = 2 μm, t = 5 μm, and d = 2 μm. Assume all of the mass in the movable half of the comb drive is in the teeth (the capacitor plates that comprise the movable electrode), and that the teeth are made of polycrystalline silicon, which has a mass density ρ = 2.33 g-cm−3.Figure Comb drive accelerometer. Problem.... Get solution
2. Two point charges. Two small identical spheres have charges of +20 nC and −5 nC, respectively. (a) What is the force between them if they are apart by 10 cm? (b) The two spheres are brought into contact and then separated again by 10 cm. What is the force between them now? Get solution
3. Two suspended charges. Two small, identical, electrically charged conducting spheres of mass 2.5 g and charge +150 nC each are suspended by weightless strings of length 12 cm each, as shown in Figure. Calculate the deflection angle θ.Figure Two suspended charges. Problem.... Get solution
4. Zero force. (a) Three point charges of Q1 = +40 nC, Q2 = −20 nC, and Q3 = +10 nC are all situated on the x axis in such a way that the net force on charge Q2 due to Q1 and Q3 is equal to zero. If Q1 and Q2 are located at points (−2, 0, 0) and (0, 0, 0), respectively, what is the location of charge Q3? (b) Q3 is moved to a different position on the x axis such that the force on itself due to Q1 and Q2 is equal to zero. What is the new position of Q3? Get solution
5. Three charges. Three identical charges of charge Q are located at the vertices of an equilateral triangle of side length a. Determine the force on one of the charges due to the other two. Get solution
6. Three point charges. Three point charges of values +150 nC, +100 nC, and +200 nC are located at points (0, 0, 0), (1, 0, 0), and (0, 1, 0), respectively. (a) Find the force on each charge due to the other charges. Which charge experiences the largest force? (b) Repeat part (a) for −100 nC as the second charge. Get solution
7. Two point charges. Two point charges of +Q and −Q are located at (0, 0, 0) and at (0, 4a, 0) respectively. Find the electric field at P1(0, 0, 3a) and at P2(0, 4a, 3a). Sketch the orientations of the fields. Get solution
8. Zero field from three charges. Two point charges of +Q each are located at (10, 0) cm and (−10, 0) cm, while a third one of charge −2Q is at (0,−10) cm. Find the coordinates of the point where the electric field is zero. Get solution
9. Three point charges. Two point charges of +10 nC each are located at points (0.46, 0, 0) and (−0.46, 0, 0), respectively. (a) Where should a third point charge of +15 nC be placed such that E = 0 at point (0, 1, 0)? (b) Repeat part (a) for a third charge of −15 nC. (c) With the −15 nC charge located as you determined in part (b), is there another point at which the electric field E = 0? If so, specify this point. Get solution
10. Four point charges on a square. Four point charges of +50 nC each are located at the corners of a square of side length 10 cm located on the xy plane and centered at the origin. (a) Find and sketch the electric potential on the z axis. (b) Find and sketch the electric field on the z axis. Get solution
11. Six point charges on a hexagon. Six identical point charges of +25 nC each are situated in space at the corners of a regular hexagon whose sides are each of length 6 cm. (a) Find the electric potential at the center of the hexagon. (b) Determine the energy required to move a point charge of −25 nC from infinity to the center of the hexagon. Get solution
12. Two straight-line charges. Consider two uniformly charged wires, each of length 1 m and a total charge +100 nC, with their ends separated by 1 m, as shown in Figure. (a) Find the electric potential ϕ at the point P, midway between the two wires. (b) Find the electric field E at point P.Figure Two straight-line charges. Problem.... Get solution
13. Seven point charges on a cube. Seven identical point charges of +10 nC each occupy seven of the eight corners of a cube 3 cm on each side. Find the electric potential at the unoccupied corner. Get solution
14. Two line charges. A uniform line charge of ρl1 = −4π × 8.85 pC-m−1 is located between the points (−5, 0) and (−2, 0) m and another such line of positive charge (i.e., ρl2 = 4π × 8.85 pC-m−1) between (5, 0) and (2, 0) m, as shown in Figure. (a) Find the electric potential ϕ at (1, 0) m. (b) Find the electric field E at the same point. At which points, if any, is the electric field E zero? At which points, if any, is the electric potential ϕ zero?Figure Two line charges. Problem.... Get solution
15. Circular ring of charge. A total charge of Q1 is distributed uniformly along a half-circular ring as shown in Figure. Two point charges, each of magnitude Q2, are situated as shown. The surrounding medium is free space. (a) Find Q2 in terms of Q1 so that the potential ϕ at the center of the ring is zero. (b) Find Q2 in terms of Q1 so that the electric field E at the center of the ring is zero.Figure Circular ring of charge. Problem.... Get solution
16. Semicircular line charge. A thin line charge of density ρl is in the form of a semicircle of radius a lying on the xy plane with its center located at the origin, as shown in Figure. Find the electric field at the origin for the cases in which (a) the line charge density ρl = ρ0 is a constant and (b) the line charge density varies along the semicircular ring as ρl = ρ0 sin φ.Figure Semicircular line charge. Problem.... Get solution
17. Charge on a hemisphere. The curved surface of a hemisphere of radius a centered at the origin carries a total charge of Q uniformly distributed over its curved surface, as shown in Figure. (a) Find the electric potential on the z axis. (b) Find the electric field on the z axis. (c) Repeat parts (a) and (b) if the charge Q is uniformly distributed throughout the volume of the hemisphere.Figure Charge on a hemisphere. Problem.... Get solution
18. Sheet of charge with hole. An infinite sheet of uniform charge density ρs is situated coincident with the xy plane at z = 0. The sheet has a hole of radius a centered at the origin. Find (a) the electric potential ϕ and (b) the electric field E at points along the z axis. Get solution
19. Spherical charge distribution. A charge density of...where K and b are constants, exists in a spherical region of space defined by 0r a. (a) Find the total charge in the spherical region. (b) Find the electric field at all points in space. (c) Find the electric potential at all points in space. (d) Show that the potential found in part (c) satisfies the equation ∇2 = ϕ−ρ(r)/∊ 0 for both r a and r >a. Get solution
20. The electron charge density in a hydrogen atom. According to quantum mechanics, the electron charge of a hydrogen atom in its ground state is distributed like a cloud surrounding its nucleus, extending in all directions with steadily decreasing density such that the total charge in this cloud is equal to qe (i.e., the charge of an electron). This electron charge distribution is given by...where a is the Bohr radius, a ≃ 0.529 × 10−10 m. (a) Find the electric potential and the electric field due to the electron cloud only. (b) Find the total electric potential and the electric field in the atom, assuming that the nucleus (proton) is localized at the origin. Get solution
21. Spherical shell of charge. The space between two concentric spheres of radii a and b (a b) in free space is charged to a volume charge density given by...where K is a constant. (a) Find the total charge in the shell. (b) Find the electric field at all points in space. (c) Find the electric potential at all points in space. (d) What happens if b → a? Get solution
22. Space charge between parallel plates. The space between two perfectly conducting parallel plates is filled with space charge (i.e., free charge) of density given as:...where d is the separation distance of the plates. Other than the space charge, the medium between the plates is air, with permittivity ∊0. The upper and lower plates, located at z = 0 and z = d are kept at potentials of ϕ = 0 and = V0, respectively. Determine expressions for the potential (z ) and the electric field E(z ) between the plates. Get solution
23. Spherical charge distribution. A spherical charge distribution exists in free space in the region 0r a given by...(a) Find the total charge. (b) Determine E everywhere. (c) Determine ϕ everywhere. (d) Sketch both |E| and ϕ as a function of r. Get solution
24. Spherical charge with a cavity. A spherical region of radius b in free space is uniformly charged with a charge density of ρ = K, where K is a constant. The sphere contains an uncharged spherical cavity of radius a. The centers of the two spheres are separated by a distance d such that d + a b. Find the electric field inside the cavity. Get solution
25. Charge on a hollow metal sphere. A hollow metal sphere of 20 cm diameter is given a total charge of 1 μC. Find the electric field and the electric potential at the center of the sphere. Get solution
26. Electric field. An electric field in empty space is given as:...Determine the energy required to move a positive unit test charge in the presence of this electric field from the point (0, 0) to (1, 1) along the shortest straight line in the x-y plane.Determine the energy required to move the same test charge from point (1, 1) back to (0, 0) but via a different path, by first going fro (1, 1) to (1, 0) and then from (1, 0) to (0, 0). Get solution
27. A 1-farad capacitor. To get an idea about the physical size of a 1-F capacitor, consider a parallel-plate capacitor with the two metal plates separated by 1 mm thickness of air. Calculate the area of the metal plates needed so that the capacitance is 1 F. Get solution
28. Gate oxide capacitance of a MOS transistor. A basic MOS transistor consists of a gate conductor and a semiconductor (which is the other conductor), separated by a gate dielectric.Consider a MOS transistor using silicon dioxide (SiO2) (∊r = 3.9) as the gate oxide. The gate oxide capacitance can be approximated as a parallel-plate capacitor. The gate oxide capacitance per unit area is given by...where ∊ox and tox are the permittivity and the thickness of the gate dielectric. (a) If the thickness of the SiO2 layer is 2 × 10−6 cm, find the gate oxide capacitance per unit area.(b) If the length and the width of the gate region are L = 5 × 10−4 cm and W = 2 × 10−3 cm, respectively, find the total gate capacitance. Get solution
29. RG 6 coaxial cable. A coaxial cable (RG 6) designed for interior use, such as connecting a TV set to a VCR, has a per-unit-length capacitance listed as 17.5 pF/ft. If the relative dielectric constant of the insulator material in the cable is ∊r ≃ 1.64, find the ratio of the inner and outer radii of the insulator. Get solution
30. Radius of a high-voltage conductor sphere. Consider an isolated charged metallic conductor sphere in a dielectric medium at an electric potential of 500 kV. Calculate the minimum radius of the sphere such that dielectric breakdown will not occur if the surrounding dielectric is (a) air (EBR =3 MV-m−1), (b) a gaseous dielectric such as sulphur hexafluoride (SF6) (∊r ≃ 1 and EBR = 7.5 MV-m−1), (c) a liquid dielectric such as oil (∊r = 2.3 and EBR = 15 MV-m−1), and (d) a solid dielectric such as mica (∊r = 5.4 and EBR = 200 MV-m−1). Get solution
31. Parallel-plate capacitor. A parallel-plate capacitor is constructed from two aluminum foils of 1 cm2 area each placed on both sides of rubber (∊r = 2.5 and EBR = 25 MV-m−1) of thickness 2.5 mm. Find the voltage rating of the capacitor using a safety factor of 10. Get solution
32. Energy in a capacitor. A 9-V battery is connected across a parallel-plate air-filled capacitor. The battery is subsequently removed, and a block of solid dielectric (∊ = 2∊0) is inserted between the plates. (a) What is the voltage across the capacitor after the introduction of the dielectric? (b) Compare the total electrostatic energy stored in this capacitor before and after the introduction of the dielectric. Comment and explain any differences. Neglect all fringing effects. Get solution
33. Coaxial capacitor. Consider a coaxial capacitor as shown in Figure. Given a = 5 mm, l = 3 cm, and the voltage rating of the capacitor to be 2 kV with a safety factor of 10, what is the maximum capacitance that can be designed using (a) oil (∊r = 2.3 and EBR = 15 MVm −1) and (b) mica (∊r = 5.4 and EBR = 200 MV-m−1).Figure Coaxial capacitor. Problem.... Get solution
34. Coaxial capacitor with two dielectrics. A coaxial capacitor consists of two conducting coaxial surfaces of radii a and b (a b). The space between is filled with two different dielectric materials with relative dielectric constants ∊1r and ∊2r , as shown in Figure. (a) Find the capacitance of this configuration. (b) Assuming that l = 5 cm, b = 3a = 1.5 cm, and oil and mica are used, calculate the capacitance. (c) Redo part (b) assuming that only oil is used throughout.Figure Coaxial capacitor with two dielectrics. Problem.... Get solution
35. Capacitor with spacers. The cross-sectional view of an air-filled coaxial capacitor with spacers made out of material with permittivity ∊ is shown in Figure. (a) Find the capacitance of this coaxial line in terms of∊, a, b, and φ. (b) If the spacers are to be made out of mica (∊ = 6∊0), determine the angle φ such that only 10% of the total energy stored by the capacitor is stored in the spacers. (c) Consider the capacitor without the spacers (i.e., φ = 0). For a given potential difference V0 between the inner and outer conductors and for a given fixed value of b, determine the inner radius a for which the largest value of the electric field is a minimum.Figure Coaxial capacitor with spacers. Problem.... Get solution
36. Earth capacitor. Consider the earth to be a large conducting sphere. (a) Find its capacitance (the earth’s radius is ∼6.371 × 106 m). (b) Find the total charge and energy stored on the earth (take the electric field on the surface of the earth to be 100 V-m−1). (c) Find the maximum charge and energy that can be stored on the earth. Get solution
37. Coaxial capacitor with variable ∊. A coaxial capacitor of inner radius a and outer radius b is filled with a dielectric material whose relative permittivity varies as ∊r = 10 a/r over the region from r = a to r = b. Find the capacitance per unit length and compare with the capacitance of the same coaxial cable when filled with air. Get solution
38. Planar charge. A surface charge distribution ρs (x, z ) exists on the x-z plane, with no charge anywhere else (i.e., ρ = 0 for |y| >0). Which of the following potential functions are valid solutions for the electrostatic potential in the half-space y >0, and what is the corresponding charge distribution ρs (x, z ) on the x-z plane?... Get solution
39. Parallel power lines. An isolated pair of parallel power lines a distance of d1 apart have a potential difference of VAB and are located a distance h above a pair of telephone wires, as shown in Figure. The parameter values are d1 = 1 m, a = 2 cm, VAB = 440 V, h = 60 cm, and d2 = 15 cm. (a) Find the direction and magnitude of the electric field at points 1 and 2. Take the midpoint between the power lines as the origin of your coordinate system. (b) Determine the potential difference ϕ12 between points 1 and 2.Figure Parallel power lines above telephone lines. Problem.... Get solution
40. Field under high-voltage line. Many 60 Hz high-voltage transmission lines operate at an rms alternating voltage of 765 kV. (a) What is the peak electric field at ground level under such a line if the wire is 12 m above the ground? (b) What is the peak potential difference between the head and feet of a 6-ft tall person? (c) Is the field sufficient to ignite a standard (110 V) fluorescent lamp of 2 ft length? Get solution
41. Capacitance of the two-wire line with dielectric sleeve. A two-wire line consists of two metallic conductors of radius a enclosed by dielectric (permittivity _) sleeves of radii b separated by a distance d as shown in Figure, with d a. The space surrounding the dielectric sleeves is air, with permittivity _0. (a) Determine the capacitance per unit length of this two-wire line. (b) The values of wire radius and separation distance are given to be a = 1 cm and d = 20 cm, respectively, while the radius of the dielectric sleeve is b = 5 cm.If the dielectric sleeve is made of mica (∊ = 5.4∊0, EBRmica = 200 MV-m–1), determine the maximum operating voltage of the two-wire line capacitor. Note that the breakdown field for air is EairBR = 3 MV-m−1.Figure Two-wire transmission line. Problem.... Get solution
42. Thundercloud fields. A typical thundercloud can be modeled as a capacitor with horizontal plates with 10 km2 area separated by a vertical distance of 5 km. Just before a large lightning discharge, the upper plate may have a total positive charge of up to 300 C, with the lower plate having an equal amount of negative charge. (a) Find the electrostatic energy stored in the cloud just before a discharge. (b) What is the potential difference between the top and bottom plates? (c) What is the average electric field within the cloud? How does this value compare to the dielectric breakdown field of dry air (3 MV-m−1)? Get solution
43. Two conducting spheres. Consider a pair of small conducting spheres with radii a, b, small compared with the separation distance d between their centers (i.e., a, b ≪ d). (a) Determine the electrostatic energy stored by this configuration, assuming that the spheres with radii a and b carry charges of Q and −Q, respectively. Your answer should depend on d. State all assumptions. (b) Repeat part (a) assuming that the spheres with radii a and b carry charges of +Q and +2Q, respectively. Get solution
44. Voltage-controlled actuator. Consider a voltage-controlled parallel-plate actuator, as shown in Figure, with plate area A = 500 × 500μm2, equilibrium separation distance d0 = 2 μm, and effective spring constant k = 0.01 N-m−1. The capacitor is filled with free space. (a) What is the usable voltage range of this actuator? (b) What is the spacing d between the top and bottom capacitor plates when a voltage of 0.05 V is applied? Get solution
45. Air-bag accelerometer. The comb-drive accelerometer depicted in Figure is to be used in an integrated circuit that is responsible for deploying an air-bag in the event of an automobile crash. Assuming a sufficient overlap between the plates such that the effects of fringing fields remain unchanged and effective spring constant k = 0.01 N-m−1, determine the change in capacitance during a deceleration of 40g, where g ≈ 9.8 m-s−2. Assume the capacitor is filled with free space, there are N = 100 capacitor plate pairs, and the following dimensional parameters: L = 200 μm, w = 2 μm, t = 5 μm, and d = 2 μm. Assume all of the mass in the movable half of the comb drive is in the teeth (the capacitor plates that comprise the movable electrode), and that the teeth are made of polycrystalline silicon, which has a mass density ρ = 2.33 g-cm−3.Figure Comb drive accelerometer. Problem.... Get solution
