Questions and Answers of Electrodynamics

(a) How long does it take a radio signal to travel 150 km from a transmitter to a receiving antenna?(b) We see a full Moon by reflected sunlight how much earlier did the light that enters our eye
In about A.D. 150, Claudius Ptolemy gave the following measured values for the angle of incidence θ1 and the angle of refraction θ2 for a light beam passing from air to water: Assuming these data
The electric component of a beam of polarized light is Ey = (5.00 V/m) sin [(l.00 x 106 m-1) z + wt].(a) Write an expression for the magnetic field component of the wave, including a value for ar.
The magnetic component of an electromagnetic wave in vacuum has amplitude of 85.8nT and an angular wave number of 4.00 m-1. What are?(a) The frequency of the wave,(b) The rms value of the electric
A helium-neon laser, radiating at 632.8 nm, has a power output of 3.0mW. The beam diverges (spreads) at angle Î¸ = 0.17mrad (Figure).(a) What is the intensity of the beam 40 m from the
The average intensity of the solar radiation that strikes normally on a surface just outside Earth's atmosphere is 1.4kW/m2.(a) What radiation pressure p, is exerted on this surface, assuming
During a test, a NATO surveillance radar system, operating at 12 GHz at 180 kW of power, attempts to detect an incoming stealth aircraft at 90 km. Assume that the radar beam is emitted uniformly over
An unpolarized beam of light is sent into a stack of four polarizing sheets, oriented so that the angle between the polarizing directions of adjacent sheets is 30°. What fraction of the incident
A beam of initially unpolarized light is sent through two polarizing sheets placed one on top of the other. What must be the angle between the polarizing directions of the sheets if the intensity of
An electromagnetic wave is traveling in the negative direction of a y axis. At a particular position and time, the electric field is directed along the positive direction of the z axis and has a
Calculate the(a) Upper and(b) Lower limit of the Brewster angle for white light incident on fused quartz. Assume that the wavelength limits of the light are 400 and 700 nm.
A particle in the solar system is under the combined influence of the Sun's gravitational attraction and the radiation force due to the Sun's rays. Assume that the particle is a sphere of density 1.0
The magnetic component of a polarized wave of light is B = (4.0 x 10-6 T) sin [(1.57 x 107 m-1) y + wt].(a) Parallel to which axis is the light polarized? What are the(b) Frequency and(c) Intensity
Three polarizing sheets are stacked. The first and third are crossed; the one between has its polarizing direction at 45.0° to the polarizing directions of the other two. What fraction of the
A ray of white light traveling through fused quartz is incident al a quartz-air interface at angle θ1. Assume that the index of refraction of quartz is n = 1.456 at the red end of the visible range
In Figure, unpolarized light is sent into the system of three polarizing sheets, where the polarizing directions of the first and third sheets are at angles θ1 = 30? (counterclockwise) and θ3 = 30?
In a region of space where gravitational forces can be neglected, a sphere is accelerated by a uniform light beam of intensity 6.0mW/m2. The sphere is totally absorbing and has a radius of 2.0 μm
In Figure, unpolarized light is sent into a system of three polarizing sheets, where the polarizing directions of the first and second sheets are at angles θ1 = 20? and θ2 = 40?. What fraction of
In Figure, unpolarized light is sent into a system of three polarizing sheets with polarizing directions at angles θ1 = 20?, θ2 = 60?, and θ3 = 40?. What fraction of the initial light intensity
A square, perfectly reflecting surface is oriented in space to be perpendicular to the light rays from the Sun. The surface has an edge length of 2.0 m and is located 3.0 x 1011 m from the Sun's
The rms value of the electric field in a certain light wave is 0.200 V/m. What is the amplitude of the associated magnetic field?
In Figure, an albatross glides at a constant 15 m/s horizontally above level ground, moving in a vertical plane that contains the Sun. It glides toward a wall of height h = 2.0 m, which it will just
The magnetic component of a polarized wave of light is given by Bx = (4.00μT) sin [k y + (2.00 x 1015 s-1t]. (a) In which direction does the wave travel? (b) Parallel to which axis is it
In Figure, where n1= 1.70, n2= 1.50, and n3= 1.30 light refracts from material 1 into material 2. If it is incident at point A at the critical angle for the interface between materials 2 and 3, what
When red light in vacuum is incident at the Brewster angle on a certain glass slab, the angle of refraction is 32.0. What are?(a) The index of refraction of the glass and(b) The Brewster angle?
Start from Eqs 33-11 and 33-17 and show that E(x, t) and B(x, t), the electric and magnetic field components of a plane traveling electromagnetic wave, must satisfy the "wave equations"
(a) Show that Eqs.33-1, and 33-2 satisfy the wave equations displayed in Problem 102.(b) Show that any expressions of the form E = Emf (k x + wt) and B = B mf (k x + wt), where f (k x + wt) denotes
A point source of light emits isotropic ally with a power of 200 W. What is the force due to the light on a totally absorbing sphere of radius 2.0 cm at a distance of 20 m from the source?
Find the unit vector along the line joining point (2, 4, 4) to point (– 3, 2, 2).
Let A = 2ax + 5ay, – 3az, B = 3ax - 4ay, and C = ax + ay + az. (a) Determine A + 2B. (b) Calculate |A – 5C|.(c) For what values of k is |kB| = 2? (d) Find (A X B) / (A
If the position vectors of points T and S are 3ax – 2ay, + az and 4ax + 4 + 6ay + 2ax, respectively, find: (a) The coordinates of T and S, (b) The distance
Given vectors T = 2ax – 6ay + 3az and S = ax + 2ay + az, find: (a) The scalar projection of T on S, (b) The vector projection of S on T, (c) The smaller angle between T and S.
If A = – ax + 6ay + 5az and B = ax + 2ay + 3ax, find: (a) The scalar projections of A on B, (b) The vector projection of B on A, (c) The unit vector perpendicular to the plane containing A and B.
Calculate the angles that vector H = 3ax + 5ay - 8az makes with the x-,y-, and z-axes.
Find the triple scalar product of P, Q, and R given thatP = 2ax - ay + azQ = ax + ay + azandR = 2ax, + 3az
Simplify the following expressions:(a) A × (A × B)(b) A × [A × (A × B)]
Show that the dot and cross in the triple scalar product may be interchanged, i.e., A ∙ (B × C) = (A × B) ∙ C.
Points P1(l, 2, 3), P2(–5, 2, 0), and P3(2, 7, –3) form a triangle in space. Calculate the area of the triangle.
The vertices of a triangle are located at (4, 1, –3), (– 2, 5, 4), and (0, 1, 6). Find the three angles of the triangle.
Points P, Q, and R are located at (– 1, 4, 8), (2, – 1, 3), and (– 1, 2, 3), respectively. Determine: (a) The distance between P and Q, (b) The distance vector from P to R, (c) The angle
If r is the position vector of the point (x, y, z) and A is a constant vector, show that: (a) (r – A) ∙ A = 0 is the equation of a constant plane (b) (r – A) ∙ r = 0 is the equation
(a) Prove that P = cos θ1 ax + sin θ1 ay and Q = cos θ2 ax + sin θ2 ay are unit vectors in the xy-plane respectively making angles θ1 and θ2 with the x-axis. (b) By
Consider a rigid body rotating with a constant angular velocity w radians per second about a fixed axis through O?as in Figure. Let r be the distance vector from O?to P,?the position of a particle in
Given A = x2yax — yzay + yz2az, determine:(a) The magnitude of A at point T(2, –1,3)(b) The distance vector from T to 5 if S is 5.6 units away from T and in the same direction as A at T(c) The
E and F are vector fields given by E = 2xax + ay + yzaz and F = xyax — y2ay+ xyzaz. Determine:(a) |E| a t (l, 2, 3)(b) The component of E along F at (1, 2, 3)(c) A vector perpendicular to both E
Using the differential length dl, find the length of each of the following curves: (a) ρ = 3, π/4 < Φ < π/2, z = constant (b) r = 1, θ = 30°, 0 < Φ < 60° (c) r = 4,
Calculate the areas of the following surfaces using the differential surface area dS: (a) ρ = 2, 0 < z < 5, π/3 < Φ < π/2 (b) z = 1, 1 < ρ < 3, 0 < Φ < π/4 (c) r
Use the differential volume dv to determine the volumes of the following regions: (a) 0 < x < 1, 1 < y < 2, - 3 < z < 3 (b) 2 < p < 5, π/3 < Φ < π, - 1 < z < 4 (c) 1 < r < 3,
Given that ps = x2 + xy, calculate ∫S ρsdS over the region y ≤ x2, 0 < x < 1.
Given that H = x2 ax + y2ay, evaluate ∫L H ∙ dl, where L is along the curve y = x2 from (0, 0) to (1, 1).
Find the volume cut from the sphere radius r = a by the cone θ = α. Calculate the volume when α = π/3 and a = π /2.
Let A = 2xyax + xzay - yaz. Evaluate ∫ A dv over:(a) A rectangular region 0 < x < 2, 0 < y < 2, 0 < z < 2(b) A spherical region r < 4
The acceleration of a particle is given by a = 2.4aτ m/s2. The initial position of the particle is r = (0, 0, 0), while its initial velocity is v = — 2ax + 5az m/s. (a) Find the position of
Find the gradient of the these scalar fields: (a) U = 4xz2 + 3yz (b) W = 2p(z2 + 1) cos Φ (c) H = r2 cos θ cos Φ
Determine the gradient of the following fields and compute its value at the specifiedpoint.
Determine the unit vector normal to S(x, y, z) — x2 + y2 – z at point (1, 3, 0).
The temperature in an auditorium is given by T = x2 + y2 – z. A mosquito located at (1, 1, 2) in the auditorium desires to fly in such a direction that it will get warm as soon as possible. In what
Find the divergence and curl of the following vectors:
The heat flow vector H = k?T, where T is the temperature and k is the thermal conductivity. Show that where then ? ? H = 0.
If r = xax + yay + zaz and T = 2zyax + xy2ay + x2yzaz, determine
If r = xax + yay + zaz is the position vector of point (x, y, z), r = |r|, and n is an integer, show that:
If r and r are as defined in the previous problem, provethat:
For each of the following scalar fields, find ?2V
Find the Laplacian of the following scalar fields and compute the value at the specifiedpoint.
If V = x2y2z2 and A = x2y ax + xz3 ay - y2z2 az, find: (a) V2V, (b) V2A, (c) Grad div A, (d) Curl curl A.
Given that F = x2y ax – y ay, find (a) ∫L F ∙ dl where L is shown in figure. (b) ∫ s (∆ x F) ∙ dS where S is the area bounded by L. (c) Is Stokes's theorem
Let D = 2pz2ap + ? cos2 ?az. Evaluateover the region defined by 0
If F = x2ax + y2ay + (z2 – 1) az, find ∫S F ∙ dS, where S is defined by p = 2, 0 < z < 2, 0 < Φ < 2, 0 < Φ < 2π.
(a) Given that A = xyax + yzay + xzaz, evaluate ∫S A ∙ dS, where S is the surface of the cube defined b y 0 < x < l, 0 < y < 1, 0 < z < 1. (b) Repeat part (a) if S remains the same but A
Verify the divergence theorem for each of the following cases: (a) A = xy2ax + y3ay + y2zaz and S is the surface of the cuboid defined by 0 (b) A = 2pzap + 3z sin 4a? ??? 4p cos ? az and S is the
The moment of inertia about the z-axis of a rigid body is proportional toExpress this as the flux of some vector field A through the surface of thebody.
Let A = ? sin ? ap + ?2 a?. Evaluate ?L A ? dl given that(a) L is the contour of Figure (a)(b) L is the contour of Figure (b)
Calculate the total outward flux of vector F = ρ2 sin Φ aρ + z cos Φ aΦ + ρzaz through the hollow cylinder defined b y 2 < p < 3, 0 < z < 5.
A rigid body spins about a fixed axis through its center with angular velocity to. If u is the velocity at any point in the body, show that w = ½ ∆ x u.
Let U and V be scalar fields, show that
Given the vector field G = (16xy – z)ax + 8x2ay – xaz (a) Is G irrotational (or conservative)? (b) Find the flux of G over the cube 0 < x, y, z < 1. (c) Determine the circulation of G around
If the vector field T = (αxy + βz3)ax + (3x2 – γz)ay + (3xz2 – y)az Is irrotational, determine α, β, and γ. Find ∆ ∙ T at (2, – 1, 0).
The magnetic vector potential at point P(r, ?, ?) due to a small antenna located at the origin is given byWhere r2 = x2 + y2 + z2. Find E(r, ?,?, t) and H(r, ?, ?, i) at the far field
A Hertzian dipole at the origin in free space has dℓ = 20 cm and 7 = 1 0 cos 2πl07t A , find |Eθs| at the distant point (100, 0, 0) .
A 2-A source operating at 300 MHz feeds a Hertzian dipole of length 5 mm situated at the origin. Find Es and Hs. at (10, 30°, 90°).
(a) Instead of a constant current distribution assumed for the short dipole of Section 13.2, assume a triangular current distribution src="/images3/39-P-E-W-A(168)-1.PNG" alt = "2|z| 1, = 1, (1 "
An antenna can be modeled as an electric dipole of length 5 m at 3 MHz. Find the radiation resistance of the antenna assuming a uniform current over its length.
A half-wave dipole fed by a 50-Ω transmission line, calculate the reflection coefficient and the standing wave ratio.
A 1-m-long car radio antenna operates in the AM frequency of 1.5 MHz. How much current is required to transmit 4 W of power?
(a) Show that the generated far field expressions for a thin dipole of length ? carrying sinusoidal current Io cos ?z are (b) On a polar coordinate sheet, plot f(?) in part (a) for ? = ?, 3?/2 and
For Problem 13.4(a) Determine Es and Hs at the far field(b) Calculate the directivity of the dipole
An antenna located on the surface of a flat earth transmits an average power of 200 kW. Assuming that all the power is radiated uniformly over the surface of a hemisphere with the antenna at the
A 100-turn loop antenna of radius 20 cm operating at 10 MHz in air is to give a 50 mV/m field strength at a distance 3 m from the loop. Determine(a) The current that must be fed to the antenna(b) The
Sketch the normalized E-field and H-field patterns for (a) A half-wave dipole (b) A quarter-wave monopole
Based on the result of Problem 13.8, plot the vertical field patterns of monopole antennas of lengths ℓ = 3λ/2, λ, 5λ/8. Note that a 5λ/8 monopole is often used in practice.
In free space, an antenna has a far-zone field given byWhere ? = ???o?o. Determine the radiated power.
At the far field, the electric field produced by an antenna is Es = — 10/r e-jβr cos θ cos Φ az Sketch the vertical pattern of the antenna. Your plot should include as many points
For an Hertzian dipole, show that the time-average power density is related to the radiation power accordingto
At the far field, an antenna produces0 Calculate the directive gain and the directivity of the antenna.
From Problem 13.8, show that the normalized field pattern of a full-wave (? = ?) antenna is given bySketch the field pattern.
For a thin dipole λ/16 long, find (a) The directive gain, (b) The directivity, (c) The effective area, (d) The radiation resistance.
Repeat Problem 13.19 for a circular thin loop antenna λ/12 in diameter.For a thin dipole A/16 long, find:(a) The directive gain,(b) The directivity,(c) The effective area,(d) The radiation
Express the following points in Cartesian coordinates: (a) P(1, 60°, 2) (b) Q(2, 90°, – 4) (c)R(, 45°, 210°) (d) T(4, π/2, π/6)
Express the following points in cylindrical and spherical coordinates:P(1, – 4 , –3)
(a) If V = xz — xy + yz, express V in cylindrical coordinates,(b) If U = x2 + 2y2 + 3z2, express U in spherical coordinates.

Showing 600 - 700 of 1324