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- Question : 1.1 - Three unequal charges in a triangle. Repeat Example 1.1 but assuming that one of the three charges in Fig. 1.3(a) amounts to (a) 3Q and (b)
- Question : 1.2 - Three charges in equilibrium. The distance between point charges Q\ = 36 pC and Qi = 9 pC is D = 3 cm. If the third charge, O3 , is placed at the line connecting Q\ and Q2 , at a distance d from Qi, as shown in Fig. 1.50, find 03 and d which ensure that all the charges in this system are in the electrostatic equilibrium, i.e., that the resultant Coulomb force on each charge is zero. Qi Q3 Qi D
- Question : 1.3 - Four charges at rectangle vertices. Four small charged bodies of equal charges Q =
- Question : 1.4 - Five charges in equilibrium. Four small charged balls of equal charges Q\
- Question : 1.5 - Three point charges in space, (a) For the three charges from Example 1.3, find the resultant electric force on the charge Q2 (Fe2 ). (b) Determine the force F e3 (on Q3). (c) What is the sum of all the three forces, Fe i + Fe2 + Fe3 ?
- Question : 1.6 - Five charges at pyramid vertices. Four point charges Q are positioned in air at the corners of the square base of a pyramid. A fifth charge
- Question : 1.7 - Eight charges at cube vertices. Eight small charged bodies of equal charges Q exist at the vertices of a cube with sides of length a, in free space. Find the magnitude and direction of the electric force on one of the charges.
- Question : 1.8 - Electric field due to three point charges in space. For the three charges from Example 1.3, determine the magnitude and direction of the electric field intensity vector at (a) the coordinate origin and (b) the point at the z-axis defined by z = 100 m.
- Question : 1.9 - Nonuniform volume charge in a cylinder. An infinitely long cylinder of radius a in free space is charged with a volume charge density p{r)
- Question : 1.10 - Nonuniform volume charge in a cube. A cube of edge length a in free space is charged over its volume with a charge density p(x) = Po sin(7TJc/a), 0 < x < a, where po is a constant and x is the normal distance from one of the cube sides. Compute the total charge of the cube
- Question : 1.11 - Nonuniform volume charge in a cube. A cube of edge length a in free space is charged over its volume with a charge density p(x) = Po sin(7TJc/a), 0 < x < a, where po is a constant and x is the normal distance from one of the cube sides. Compute the total charge of the cube
- Question : 1.12 - Nonuniform line charge along a rod. A rod of length / in air is charged with a line charge of density Q'(x) = Q' q [ 1
- Question : 1.13 - Field maximum at the axis of a ring, (a) For the charged ring in Fig. 1.11, assume Q > 0, and find z for which the electric field intensity along the z-axis is maximum, (b) Plot the function E z (z),
- Question : 1.14 - Point charge equivalent to a charged semicircle. Show that far away along the z-axis, the semicircular line charge in Fig. 1.12(a) is equivalent to a point charge with the same amount of charge located at the coordinate origin.
- Question : 1.15 - Charged contour of complex shape. Fig. 1.51 shows a contour consisting of two semicircular parts, of radii a and b (a < b ), and two linear parts, each of length b
- Question : 1.16 - Nonuniform line charge along a semicircle. Consider the geometry in Fig. 1.12(a), and assume that the charge along the semicircle is nonuniform, given by Q\4>) = Q' () s\n4> (
- Question : 1.17 - Line charge along three-quarters of a circle. A uniform line charge in the form of an arc that is 3/4 of a circle of radius a is situated in air. The total charge of the arc is Q. Calculate the electric field intensity vector at the arc center.
- Question : 1.18 - Line charge along a quarter of a circle. A charge of density Q' in free space is distributed uniformly along an arc representing a quarter of a circle of radius a. Determine the electric field intensity vector at an arbitrary point along the axis that contains the arc center and is normal to the arc plane.
- Question : 1.19 - Semi-infinite line charge. A line charge of uniform charge density Q' is distributed in free space along the negative part of the x-axis in the Cartesian coordinate system (
- Question : 1.20 - Half-positive, half-negative infinite line charge. A line charge in free space is distributed along the x-axis in the Cartesian coordinate system. The line charge density is Q' (Q' > 0) for
- Question : 1.21 - Half-positive, half-negative infinite line charge. A line charge in free space is distributed along the x-axis in the Cartesian coordinate system. The line charge density is Q' (Q' > 0) for
- Question : 1.22 - Point charge equivalent to a charged disk. Consider the charged disk in Fig. 1.14, and show that for |z| a, the E field in Eq. (1.63) is equivalent to the field of a point charge Q = ps 7ta2 placed at the disk center.
- Question : 1.23 - Field due to a nonuniformly charged disk. Consider the disk with a nonuniform charge distribution from Problem 1.11, and find the electric field intensity vector along the disk axis normal to its plane.
- Question : 1.24 - Nonuniformly charged spherical surface. A sphere of radius a in free space is nonuniformly charged over its surface such that the charge density is given by ps (0) = pso sin 20, where pso is a constant and the angle 0 (0 < 0 < n) is defined as in Fig. 1.10 or 1.16. Compute (a) the total charge of the sphere and (b) the electric field intensity vector at the sphere center.
- Question : 1.25 - Infinite charged sheet with a circular hole. An infinite sheet of charge with a constant density ps has a hole of radius a in it. The sheet is in the xy-plane of the Cartesian coordinate system and the center of the hole is at the coordinate origin. The ambient medium is air. Under these circumstances, determine the electric field intensity vector at an arbitrary point along the z-axis - in the following two ways, respectively: (a) integrating the fields due to elementary rings as in Fig. 1.14 and (b) combining the results of Examples 1.11 (infinite sheet of charge, with no hole) and 1.10 (charged disk).
- Question : 1.26 - Force on a charged semicylinder due to a line charge. For the structure composed from a line charge and a charged semicylinder shown in Fig. 1.17(a) and described in Example 1.13, find the force per unit length on the semicylinder.
- Question : 1.27 - Charged strip. Consider an infinitely long uniformly charged strip of width a and surface charge density ps in air. Using the geometrical representation of the cross section of the problem as in Fig. 4.11 in Chapter 4 (also see Fig. 1.13) and change of integration variables given by Eqs. (4.43) and (4.44), obtain the expression for the E field at an arbitrary point in space due to this charge.
- Question : 1.28 - Two parallel oppositely charged strips. Two parallel, very long strips are uniformly charged with charge densities ps and
- Question : 1.29 - Work in an electrostatic field. What is the work done by electric forces in moving a charge Q = 1 nC from the coordinate origin to the point (1 m, 1 m, 1 m) in the electrostatic field given by E(x, y, z) = (x x + y2 y
- Question : 1.30 - Work in the field of a point charge. A point charge Q\ = 10 nC is positioned at the center of a square contour a = 10 cm on a side, as shown in Fig. 1.53. Find the work done by electric forces in carrying a charge Qi =
- Question : 1.31 - Electric potential due to three point charges in space. For the three charges from Example 1.3, calculate the electric potential at points defined by (a) (0,0,2 m) and (b) (1 m, 1 m, 1 m), respectively.
- Question : 1.32 - Point charge and an arbitrary reference point. Derive the expression for the potential at a distance r from a point charge Q in free space with respect to the reference point which is an arbitrary (finite) distance rji away from the charge.
- Question : 1.33 - Potential due to a semicircular line charge. Prove that the electric scalar potential at an arbitrary point along the e-axis in the field of the semicircular line charge shown in Fig. 1.12(a) and described in Example 1.7 is V = Q' a / (4eoV z 2 + a2 ).
- Question : 1.34 - Potential due to a charged disk. For the charged disk from Example 1.10, derive the following expression for the electric scalar potential along the e-axis (
- Question : 1.35 - Potential due to a hemispherical surface charge. Consider the hemispherical surface charge from Example 1.12, and find the electric scalar potential at the hemisphere center (z = 0).
- Question : 1.36 - Potential due to a nonuniform spherical surface charge. Determine the electric potential at the center of the nonuniformly charged spherical surface from Problem 1.24.
- Question : 1.37 - Voltage due to two point charges. Two point charges, Q\
- Question : 1.38 - Sketch field from potential. The electrostatic potential V in a region is a function of a single rectangular coordinate x, and V (x) is shown in Fig. 1.54. Sketch the electric field intensity E x {x) in this region.
- Question : 1.39 - Field from potential, point charge. For a point charge in free space, obtain the expression for E in Eq. (1.24) from the expression for V in Eq. (1.80).
- Question : 1.40 - Field from potential, charged semicircle. For the semicircular line charge from Example 1.7, (a) obtain the expression for Ez in Eq. (1.50) from the expression for V given in Problem 1.33 and (b) explain why it is impossible to obtain the expression for Ex in Eq. (1.48) from this same expression for V
- Question : 1.41 - Field from potential, charged disk. For the charged disk from Example 1.10, obtain the expression for E in Eq. (1.63) from the expression for V given in Problem 1.34.
- Question : 1.42 - Field from potential, charged hemisphere. For the hemispherical surface charge from Example 1.12, explain why we cannot obtain the expression for E at the hemisphere center (z = 0), given in Eq. (1.67), from the expression for V computed in Problem 1.35.
- Question : 1.43 - Angle between field lines and equipotential surfaces. Using the concept of gradient, prove that in an arbitrary electrostatic field, field lines are perpendicular to equipotential surfaces (as in Fig. 1.22).
- Question : 1.44 - Direction of the steepest ascent. The terrain elevation in a region is given by a function h(x,y) = lOOxlny [m] (x, y in km), where x and y are coordinates in the horizontal plane and 1 km < x, y < 10 km. (a) What is the direction of the steepest ascent at (3 km, 3 km)? (b) How steep, in degrees, is the ascent in (a)?
- Question : 1.45 - Maximum increase in electrostatic potential. The electrostatic field intensity vector in a region is given by E(x, y, z) = (4 x
- Question : 1.46 - Large and small electric dipole. Two point charges, Q\ = 1 nC and Q2
- Question : 1.47 - Potential and field due to a small electric dipole. An electric dipole with a moment p = 1 pCm z is located at the origin of a spherical coordinate system. The length of the dipole is d = 1 cm. Find V and E at the following points defined by spherical coordinates: (a) (1 m, 0, 0), (b) (1 m, n/2, n/2), (c) (1 m, n, 0), (d) (1 m, jt/4,0), (e) (10 m, zr/4, 0), and (f) (100 m, jr/4, 0).
- Question : 1.48 - Dipole equivalent to a nonuniform line charge. Consider the nonuniform line charge distribution along the semicircle from Problem 1.16, and show that far away along the zaxis (|z|
- Question : 1.49 - Expression for the electric field due to a line dipole. For the line dipole in Fig. 1.29, obtain the expression for E from the expression for V in Eq. (1.121).
- Question : 1.50 - Near and far potential and field due to a line dipole. Two infinite line charges, with densities Qx = 100 pC/m and Q'2 =
- Question : 1.51 - Flux of the electric field vector through a cube side. A point charge Q is located at the center of a cube in free space. The cube edges are a long. Find the outward flux of the electric field intensity vector due to this charge through each of the cube sides.
- Question : 1.52 - Flux for a different placement of the point charge. If the point charge Q from the previous problem is placed at the center of a side of the cube, determine the total outward flux of the electric field vector due to the charge through the surface composed of the remaining five sides of the cube.
- Question : 1.53 - Field of a point charge from Gauss
- Question : 1.54 - Uniformly charged thin spherical shell. An infinitely thin spherical shell of radius a in free space is uniformly charged over its surface with a total charge Q. Determine: (a) the electric field intensity vector inside and outside the shell, (b) the potential of the shell, and (c) the potential at the shell center.
- Question : 1.55 - Sphere with a nonuniform volume charge. Find the distribution of the electric scalar potential inside and outside the sphere with the volume charge density given by Eq. (1.32).
- Question : 1.56 - Field of an infinite line charge from Gauss
- Question : 1.57 - Uniformly charged thin cylindrical shell. An infinitely long and infinitely thin cylindrical shell of radius a is situated in free space. The shell is charged over its surface with a uniform
- Question : 1.58 - Cylinder with uniform volume charge. Compute the voltage between the surface and the axis of a uniformly charged infinite cylinder of radius a in free space, if the volume charge density in the cylinder is p.
- Question : 1.59 - Field of an infinite sheet of charge from Gauss
- Question : 1.60 - Two parallel oppositely charged sheets. Two parallel infinite sheets of charge with densities ps and
- Question : 1.61 - An infinitely large layer of charge in free space has a uniform volume charge density p and thickness d. (a) Compute the electric field vector inside the layer, (b) Show that, as far as the field outside the layer is concerned, the layer can be replaced by an equivalent infinite sheet of charge, and find the surface charge density, ps , of this sheet.
- Question : 1.62 - Layer with a cosine volume charge distribution. The density of a volume charge in free space depends on the Cartesian coordinate x only and is given by p{x) = pocosinx/a) (|x| < a) and p(x)
- Question : 1.63 - Layer with a sine charge distribution. Repeat the previous problem but for the following charge density function: p(x) = po sin(7rx/o) for |jc| < a (there is no charge outside the layer).
- Question : 1.64 - Exponential charge distribution in the entire space. A volume charge distribution in free space is described in the rectangular coordinate system as p(x)
- Question : 1.65 - Uniform electric field. In a certain region, there is a uniform electric field, Eo- What is the volume charge density in that region?
- Question : 1.66 - Charge distribution from 1-D field distribution. Find the volume charge density p(x) in the electrostatic system from Example 1.16, assuming that the permittivity of the medium is e0 .
- Question : 1.67 - Charge from field, planar symmetry. From the field expressions in Eqs. (1.150)
- Question : 1.68 - Charge from field, cylindrical symmetry. From the field with a radial cylindrical component only given by Eqs. (1.145) and (1.146), obtain the corresponding charge distribution in free space [Eq. (1.143)].
- Question : 1.69 - Charge from field, spherical symmetry. Using Gauss
- Question : 1.70 - Nonuniformly charged sphere using differential Gauss
- Question : 1.71 - Problem with cylindrical symmetry by differential Gauss
- Question : 1.72 - Problem with planar symmetry using differential Gauss
- Question : 1.73 - Antisymmetrical charge, differential Gauss
- Question : 1.74 - Gauss
- Question : 1.75 - Excentric charged sphere inside an uncharged shell. Consider the structure from Example 1.27, and assume that the sphere is moved toward the shell wall so that the centers of the sphere and the shell are separated by a distance d. Find the potential of the shell in the new electrostatic state if (a) d = (b
- Question : 1.76 - Point charge inside a charged shell. A point charge 2Q is placed at the center of an airfilled spherical metallic shell, charged with Q and situated in air. The inner and outer radii of the shell are a and b (a < b). (a) What is the total charge on the inner and on the outer surface of the shell, respectively? (b) Find the potential of the shell.
- Question : 1.77 - Three concentric shells, one uncharged. Three concentric spherical metallic shells are situated in air. The outer radius of the inner shell is a = 30 mm, and its charge Q
- Question : 1.78 - Three concentric shells, two at the same potential. Consider a structure with the same geometry as in the previous problem, and assume that the charges of the inner and outer shells are Q\ = 2 nC and Q3 =
- Question : 1.79 - Four coaxial cylindrical conductors. Four very long conductors, each in the form of a cylindrical shell with thickness d= 1 cm, are positioned in air coaxially with respect to each other, as indicated in Fig. 1.55, which shows a detail of the cross section of the system. The first and the fourth conductor are grounded, and the potential of the third conductor with respect to the ground is V 3 = 1 kV. The second conductor is uncharged. Find the charges per unit length of the first and the third conductor, Q\ and Figure 1.55 Detail of the cross section of a system of four cylindrical conductors; for Problem 1 .79.
- Question : 1.80 - Three concentric conductors, one grounded. Shown in Fig. 1.56 is a system consisting of three concentric spherical conductors (the inner conductor is a solid sphere, while the remaining two are spherical shells). The radius of the inner conductor is a
- Question : 1.81 - Charged metallic foil. An infinitely large flat metallic foil is situated in air and charged uniformly with the surface charge density ps = 1 nC/m2 . Find the electric field intensity vector everywhere.
- Question : 1.82 - Two metallic slabs. An infinitely large metallic slab of thickness d = 1 cm is situated in air and charged such that the surface charge density at each of the slab surfaces is ps = 1 pC/m 2 . Another metallic slab of the same thickness, which is uncharged, is then introduced and placed parallel to the charged slab such that the distance between the surfaces of the two slabs facing each other is D
- Question : 1.83 - Two metallic spheres at the same potential. Consider the system in Fig. 1.45, and assume that a
- Question : 1.84 - MoM-based computer program for a charged plate. Using the method of moments as presented in Section 1.20, write a computer program to determine the charge distribution on a very thin charged square plate of edge length a at a potential Vq, in free space. Subdivide the plate into N square patches, and assume that a = 1 m and Vq = 1 V. (a) Tabulate and plot the results for the surface charge density (ps ) of the patches, taking N = 100 (ten partitions in each dimension), (b) Compute the total charge of the plate, taking (i) N = 9, (ii) N
- Question : 1.85 - MoM computation for a charged cube. Write a computer program for the method-of-moments analysis of a charged metallic cube, Fig. 1.46, with edge length a = 1 m, and compute the total charge of the cube for Vq = 1 V and ten, or as many as possible (given available computational resources), subdivisions per cube edge ( N = 600 if ten subdivisions per edge are adopted).
- Question : 1.86 - Approximate integral expression for the electric field vector, (a) Write the approximate integral expression for the evaluation of the electric field intensity vector at an arbitrary point in space due to a charged body (e.g., the cube in Fig. 1.46), whose charge distribution is approximately described by Eq. (1.212). (b) Using the expression in (a) and the associated computer program, compute the electric field along the axis of the plate from Problem 1.84 perpendicular to its plane at points that are a/2, 2a, and 100fl, respectively, distant from the plate surface (for N
- Question : 1.87 - Force on a point charge due to its image. Find the electric force on the point charge Q in Fig. 1.48(a).
- Question : 1.88 - Imaging a line charge. For the structure defined in Example 1.30, determine the distribution of induced surface charges on the conducting plane.
- Question : 1.89 - Charged wire parallel to a corner screen. Fig. 1.57 shows a cross section of the structure consisting of a metallic wire of radius a and a 90

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