Questions and Answers of Principles Communications Systems

Let us assume that we have a very thin, square, imperfectly conducting plate 2 m on a side, located in the plane z = 0 with one corner at the origin such that it lies entirely within the first
Two point charges, 1 nC at (0, 0, 0.1) and −1 nC at (0, 0,−0.1), are in free space.(a) Calculate V at P(0.3, 0, 0.4).(b) Calculate |E| at P.(c) Now treat the two charges as a dipole at the origin
Use the electric field intensity of the dipole [Section 4.7, Eq. (35)] to find the difference in potential between points at Î¸aand Î¸b, each point having the same r and
A dipole having a moment p = 3ax − 5ay + 10az nC · m is located at Q(1, 2,−4) in free space. Find V at P(2, 3, 4).
A dipole for which p = 10∈0az C · m is located at the origin. What is the equation of the surface on which Ez = 0 but E ≠ 0?
A potential field in free space is expressed as V = 20/(xyz)V.(a) Find the total energy stored within the cube 1 < x, y, z < 2.(b) What value would be obtained by assuming a uniform energy
(a) Using Eq. (35), find the energy stored in the dipole field in the region r > a.(b) Why can we not let a approach zero as a limit? Qd 4πε0r3 + sin θ a ) . (2 cos θ a, E =
A copper sphere of radius 4 cm carries a uniformly distributed total charge of 5 μC in free space.(a) Use Gauss’s law to find D external to the sphere.(b) Calculate the total energy stored in the
A sphere of radius a contains volume charge of uniform density Ï0C/m3. Find the total stored energy by applying(a) Eq. (42);(b) Eq. (44). „V dv WE vol = +/_ D-Edv = D.E dv = , €0E²
Four 0.8 nC point charges are located in free space at the corners of a square 4 cm on a side.(a) Find the total potential energy stored.(b) A fifth 0.8 nC charge is installed at the center of the
Surface charge of uniform density Ïslies on a spherical shell of radius b, centered at the origin in free space.(a) Find the absolute potential everywhere, with zero reference at
The value of E at P(ρ = 2, ϕ = 40◦, z = 3) is given as E = 100aρ − 200aϕ + 300az V/m. Determine the incremental work required to move a 20 μC charge a distance of 6μm:(a) In the direction
Given the flux density D = 16/r cos(2θ) aθ C/m2, use two different methods to find the total charge within the region 1 < r < 2 m, 1 < θ < 2 rad, 1 < ϕ < 2 rad.
(a) Use Maxwell’s first equation, ∇ · D = ρv, to describe the variation of the electric field intensity with x in a region in which no charge density exists and in which a nonhomogeneous
In the region of free space that includes the volume 2 < x, y, z < 3, D = 2/z2 (yz ax + xz ay − 2xy az) C/m2.(a) Evaluate the volume integral side of the divergence theorem for the volume
Repeat Problem 3.8, but use ∇ ·D = ρν and take an appropriate volume integral.In ProblemUse Gauss’s law in integral form to show that an inverse distance field in spherical coordinates, D =
Let D = 5.00r2ar mC/m2 for r ≤ 0.08 m and D = 0.205 ar /r2 μC/m2 for r ≥ 0.08 m.(a) Find ρν for r = 0.06 m.(b) Find ρν for r = 0.1 m.(c) What surface charge density could be located at r =
If we have a perfect gas of mass density ρm kg/m3, and we assign a velocity U m/s to each differential element, then the mass flow rate is ρmU kg/(m2 − s). Physical reasoning then leads to the
Within the spherical shell, 3 < r < 4 m, the electric flux density is given as D = 5(r − 3)3 ar C/m2.(a) What is the volume charge density at r = 4?(b) What is the electric flux density at r
In a region in free space, electric flux density is found to beEverywhere else, D = 0.(a) Using ˆ‡ · D = Ïv, find the volume charge density as a function of position
(a) A point charge Q lies at the origin. Show that div D is zero everywhere except at the origin.(b) Replace the point charge with a uniform volume charge density ρv0 for 0 < r < a. Relate
(a) A flux density field is given as F1 = 5az. Evaluate the outward flux of F1 through the hemispherical surface, r = a, 0 < θ < π/2, 0 < ϕ < 2π.(b) What simple observation would have
Calculate ∇ · D at the point specified if(a) D = (1/z2)[10xyz ax + 5x2z ay + (2z3 − 5x2 y) az] at P(−2, 3, 5);(b) D = 5z2 aρ + 10ρz az at P(3,−45◦, 5);(c) D = 2r sin θ sin ϕ ar + r cos
A radial electric field distribution in free space is given in spherical coordinates as:where Ï0, a, and b are constants.(a) Determine the volume charge density in the entire region (0
A spherical surface of radius 3 mm is centered at P(4, 1, 5) in free space. Let D = xax C/m2. Use the results of Section 3.4 to estimate the net electric flux leaving the spherical surface.
State whether the divergence of the following vector fields is positive, negative, or zero:(a) the thermal energy flow in J/(m2 − s) at any point in a freezing ice cube;(b) the current density in
A cube is defined by 1 < x, y, z < 1.2. If D = 2x2yax + 3x2y2ay C/m2(a) Apply Gauss’s law to find the total flux leaving the closed surface of the cube.(b) Evaluate ∇ · D at the center of
An electric flux density is given by D = D0 aρ, where D0 is a given constant.(a) What charge density generates this field?(b) For the specified field, what total charge is contained within a
Volume charge density is located as follows: ρν = 0 for ρ < 1 mm and for ρ > 2 mm, ρν = 4ρ μC/m3 for 1 < ρ < 2 mm.(a) Calculate the total charge in the region 0 < ρ < ρ1,
A certain light-emitting diode (LED) is centered at the origin with its surface in the xy plane. At far distances, the LED appears as a point, but the glowing surface geometry produces a far-field
Spherical surfaces at r = 2, 4, and 6 m carry uniform surface charge densities of 20 nC/m2, −4 nC/m2, and ρS0, respectively.(a) Find D at r = 1, 3, and 5 m.(b) Determine ρS0 such that D = 0 at r
The sun radiates a total power of about 3.86 × 1026 watts (W). If we imagine the sun’s surface to be marked off in latitude and longitude and assume uniform radiation,(a) what power is radiated by
An infinitely long cylindrical dielectric of radius b contains charge within its volume of density ρv = aρ2, where a is a constant. Find the electric field strength, E, both inside and outside the
A uniform volume charge density of 80 μC/m3 is present throughout the region 8 mm < r < 10 mm. Let ρν = 0 for 0 < r < 8 mm.(a) Find the total charge inside the spherical surface r = 10
Use Gauss’s law in integral form to show that an inverse distance field in spherical coordinates, D = Aar /r , where A is a constant, requires every spherical shell of 1 m thickness to contain 4πA
Volume charge density is located in free space as ρν = 2e−1000r nC/m3 for 0 < r < 1 mm, and ρν = 0 elsewhere.(a) Find the total charge enclosed by the spherical surface r = 1 mm.(b) By
In free space, a volume charge of constant density ρν = ρ0 exists within the region −∞ < x < ∞,−∞ < y < ∞, and −d/2 < z < d/2. Find D and E everywhere.
Let D = 4xyax + 2(x2 + z2)ay + 4yzaz nC/m2 and evaluate surface integrals to find the total charge enclosed in the rectangular parallelepiped 0 < x < 2, 0 < y < 3, 0 < z < 5 m.
An electric field in free space is E = (5z3/∈0) âz V/m. Find the total charge contained within a sphere of 3-m radius, centered at the origin.
The cylindrical surface ρ = 8 cmcontains the surface charge density, ρS = 5e−20|z| nC/m2.(a) What is the total amount of charge present?(b) How much electric flux leaves the surface ρ = 8 cm, 1
An electric field in free space is E = (5z2/∈0) âz V/m. Find the total charge contained within a cube, centered at the origin, of 4-m side length, in which all sides are parallel to coordinate
Suppose that the Faraday concentric sphere experiment is performed in free space using a central charge at the origin, Q1, and with hemispheres of radius a. A second charge Q2 (this time a point
For fields that do not vary with z in cylindrical coordinates, the equations of the streamlines are obtained by solving the differential equation Ep/Eϕ = dρ/(ρdϕ). Find the equation of the line
If E = 20e−5y(cos 5xax − sin 5xay), find(a) |E| at P(π/6, 0.1, 2);(b) A unit vector in the direction of E at P;(c) The equation of the direction line passing through P.
An electric dipole (discussed in detail in Section 4.7) consists of two point charges of equal and opposite magnitude ±Q spaced by distance d. With the charges along the z axis at positions z =
Given the electric field E = (4x − 2y)ax − (2x + 4y)ay, find(a) The equation of the streamline that passes through the point P(2, 3,−4);(b) A unit vector specifying the direction of E at
A radially dependent surface charge is distributed on an infinite flat sheet in the x-y plane and is characterized in cylindrical coordinates by surface density ρs = ρ0/ρ, where ρ0 is a constant.
Find E at the origin if the following charge distributions are present in free space: point charge, 12 nC, at P(2, 0, 6); uniform line charge density, 3 nC/m, at x = −2, y = 3; uniform surface
(a) Find the electric field on the z axis produced by an annular ring of uniform surface charge density ρs in free space. The ring occupies the region z = 0, a ≤ ρ ≤ b, 0 ≤ ϕ ≤ 2π in
Given the surface charge density, ρs = 2μC/m2, existing in the region ρ < 0.2 m, z = 0, find E at(a) PA(ρ = 0, z = 0.5);(b) PB(ρ = 0, z = − 0.5). Show that(c) The field along the z axis
Two identical uniform sheet charges with ρs = 100 nC/m2 are located in free space at z = ±2.0 cm. What force per unit area does each sheet exert on the other?
Two identical uniform line charges, with ρl = 75 nC/m, are located in free space at x = 0, y = ±0.4 m. What force per unit length does each line charge exert on the other?
A line charge of uniform charge density ρ0 C/m and of length ℓ is oriented along the z axis at − ℓ/2 < z < ℓ/2.(a) Find the electric field strength, E, in magnitude and direction at
A uniform line charge of 2μC/m is located on the z axis. Find E in rectangular coordinates at P(1, 2, 3) if the charge exists from(a) −∞ < z < ∞;(b) −4 ≤ z ≤ 4.
(a) Find E in the plane z = 0 that is produced by a uniform line charge, ρL, extending along the z axis over the range −L < z < L in a cylindrical coordinate system.(b) If the finite line
A uniform line charge of 16 nC/m is located along the line defined by y = −2, z = 5. If ∈ = 0:(a) Find E at P(1, 2, 3).(b) Find E at that point in the z = 0 plane where the direction of E is
Within a region of free space, charge density is given as ρν = ρ0 rcos θ/a C/m3, where ρ0 and a are constants. Find the total charge lying within(a) The sphere, r ≤ a;(b) The cone, r ≤
A spherical volume having a 2-μm radius contains a uniform volume charge density of 1015 C/m3.(a) What total charge is enclosed in the spherical volume?(b) Now assume that a large region contains
The electron beam in a certain cathode ray tube possesses cylindrical symmetry, and the charge density is represented by ρv = −0.1/(ρ2 + 10−8) pC/m3 for 0 < ρ < 3 × 10−4 m, and ρv =
A uniform volume charge density of 0.2μC/m3 is present throughout the spherical shell extending from r = 3 cm to r = 5 cm. If ρν = 0 elsewhere, find(a) The total charge present throughout the
Electrons are in random motion in a fixed region in space. During any 1μs interval, the probability of finding an electron in a subregion of volume 10−15 m2 is 0.27. What volume charge density,
A charge Q0 located at the origin in free space produces a field for which Ez = 1 kV/m at point P(−2, 1,−1).(a) Find Q0. Find E at M(1, 6, 5) in(b) rectangular coordinates; (c) cylindrical
A charge of −1 nC is located at the origin in free space. What charge must be located at (2, 0, 0) to cause Ex to be zero at (3, 1, 1)?
A 100-nC point charge is located at A(−1, 1, 3) in free space.(a) Find the locus of all points P(x, y, z) at which Ex = 500 V/m.(b) Find y1 if P(−2, y1, 3) lies on that locus.
A 2-μC point charge is located at A(4, 3, 5) in free space. Find Eρ, Eφ, and Ez at P(8, 12, 2).
Two point charges of equal magnitude q are positioned at z = ±d/2.(a) Find the electric field everywhere on the z axis;(b) Find the electric field everywhere on the x axis;(c) Repeat parts (a) and
Let a point charge Q1 = 25 nC be located at P1(4,−2, 7) and a charge Q2 = 60 nC be at P2(−3, 4,−2).(a) If = 0, find E at P3(1, 2, 3).(b) At what point on the y axis is Ex = 0?
Eight identical point charges of Q C each are located at the corners of a cube of side length a, with one charge at the origin, and with the three nearest charges at (a, 0, 0), (0, a, 0), and (0, 0,
Point charges of 50 nC each are located at A(1, 0, 0), B(−1, 0, 0), C(0, 1, 0), and D(0,−1, 0) in free space. Find the total force on the charge at A.
Point charges of 1 nC and −2 nC are located at (0, 0, 0) and (1, 1, 1), respectively, in free space. Determine the vector force acting on each charge.
Three point charges are positioned in the x-y plane as follows: 5 nC at y = 5 cm, −10 nC at y = −5 cm, and 15 nC at x = −5 cm. Find the required x-y coordinates of a 20-nC fourth charge that
Consider a problem analogous to the varying wind velocities encountered by transcontinental aircraft. We assume a constant altitude, a plane earth, a flight along the x axis from 0 to 10 units, no
Express the unit vector ax in spherical components at the point:(a) r = 2, θ = 1 rad, ϕ = 0.8 rad;(b) x = 3, y = 2, z = −1;(c) ρ = 2.5, ϕ = 0.7 rad, z = 1.5.
State whether or not A = B and, if not, what conditions are imposed on A and B when(a) A · ax = B · ax;(b) A × ax = B × ax;(c) A · ax = B · ax and A × ax = B × ax;(d) A · C = B · C and A ×
The surfaces r = 2 and 4, θ = 30° and 50°, and ϕ = 20° and 60° identify a closed surface. Find(a) The enclosed volume;(b) The total area of the enclosing surface;(c) The total length of the
Express the uniform vector field F = 5ax in(a) Cylindrical components;(b) Spherical components.
Given point P(r = 0.8, θ = 30°, ϕ = 45°) and E = 1/r2 [cos ϕ ar + (sin ϕ/ sin θ) aϕ], find(a) E at P;(b) |E| at P;(c) A unit vector in the direction of E at P.
Two unit vectors, a1 and a2, lie in the xy plane and pass through the origin. They make angles ϕ1 and ϕ2, respectively, with the x axis(a) Express each vector in rectangular components;(b) Take the
The surfaces ρ = 3, ρ = 5, ϕ = 100°, ϕ = 130°, z = 3, and z = 4.5 define a closed surface. Find(a) The enclosed volume;(b) the total area of the enclosing surface;(c) The total length of the
A sphere of radius a, centered at the origin, rotates about the z axis at angular velocity Ωrad/s. The rotation direction is clockwise when one is looking in the positive z direction.(a) Using
Express in cylindrical components:(a) The vector from C(3, 2,−7) to D(−1, −4, 2);(b) A unit vector at D directed toward C;(c) A unit vector at D directed toward the origin.
If the three sides of a triangle are represented by vectors A, B, and C, all directed counterclockwise, show that |C|2 = (A + B) · (A + B) and expand the product to obtain the law of cosines.
(a) Express the field D = (x2 + y2)−1(xax + yay) in cylindrical components and cylindrical variables.(b) Evaluate D at the point where ρ = 2, φ = 0.2π, and z = 5, expressing the result in
A certain vector field is given as G = (y + 1)ax + xay.(a) Determine G at the point (3,−2, 4);(b) Obtain a unit vector defining the direction of G at (3,−2, 4).
Point A(−4, 2, 5) and the two vectors, RAM = (20, 18 − 10) and RAN = (−10, 8, 15), define a triangle. Find(a) A unit vector perpendicular to the triangle;(b) A unit vector in the plane of the
If A represents a vector one unit long directed due east, B represents a vector three units long directed due north, and A + B = 2C − D and 2A − B = C + 2D, determine the length and direction of
Three vectors extending from the origin are given as r1 = (7, 3,−2), r2 = (−2, 7,−3), and r3 = (0, 2, 3). Find(a) A unit vector perpendicular to both r1 and r2;(b) A unit vector perpendicular
Given that A + B + C = 0, where the three vectors represent line segments and extend from a common origin, must the three vectors be coplanar? If A + B + C + D = 0, are the four vectors coplanar?
Find(a) The vector component of F = 10ax − 6ay + 5az that is parallel to G = 0.1ax + 0.2ay + 0.3az;(b) The vector component of F that is perpendicular to G;(c) The vector component of G that is
Write an expression in rectangular components for the vector that extends from (x1, y1, z1) to (x2, y2, z2) and determine the magnitude of this vector.
Given the points M(0.1,−0.2,−0.1), N(−0.2, 0.1, 0.3), and P(0.4, 0, 0.1), find(a) The vector RMN;(b) The dot product RMN · RMP;(c) The scalar projection of RMN on RMP;(d) The angle between RMN
By expressing diagonals as vectors and using the definition of the dot product, find the smaller angle between any two diagonals of a cube, where each diagonal connects diametrically opposite corners
A field is given as G = [25/(x2+ y2)](xax+ yay). Find(a) A unit vector in the direction of G at P(3, 4,ˆ’2);(b) The angle between G and ax at P;(c) The value of the following double
Demonstrate the ambiguity that results when the cross product is used to find the angle between two vectors by finding the angle between A = 3ax − 2ay + 4az and B = 2ax + ay − 2az . Does this
Given the vector field E = 4zy2 cos 2xax + 2zy sin 2xay + y2 sin 2xaz for the region |x|, |y|, and |z| less than 2, find(a) The surfaces on which Ey = 0; (b) The region in which Ey = Ez;(c) The
Find the acute angle between the two vectors A = 2ax + ay + 3az and B = ax − 3ay + 2az by using the definition of(a) The dot product;(b) The cross product.
A vector field is specified as G = 24xyax + 12(x2 + 2)ay + 18z2az. Given two points, P(1, 2,−1) and Q(−2, 1, 3), find(a) G at P;(b) A unit vector in the direction of G at Q;(c) A unit vector
A circle, centered at the origin with a radius of 2 units, lies in the xy plane. Determine the unit vector in rectangular components that lies in the xy plane, is tangent to the circle at (−
The vector from the origin to point A is given as (6,−2,−4), and the unit vector directed from the origin toward point B is (2,−2, 1)/3. If points A and B are ten units apart, find the
Vector A extends from the origin to (1, 2, 3), and vector B extends from the origin to (2, 3,−2). Find(a) the unit vector in the direction of (A − B);(b) the unit vector in the direction of the

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