Introduction to Electrodynamics
4th Edition
ISBN: 9781108420419
Author: David J. Griffiths
Publisher: Cambridge University Press
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Question 2
a. Electrostatic fields have many applications, but most are low “low-power"
applications; that is relatively low forces are involved. Explain why this is so.
b. Given that D= (10ră ) ar (c/m²) in cylindrical coordinates, evaluate both sides of the
divergence theorem for the volume enclosed by r= Im, r= 2m and Z= 0 and Z= 10
Problem 2.20 One of these is an impossible electrostatic field. Which one?
(a) E =k[xyÂ+2yzý+3xz2];
(b) E = k[y² + (2xy + z²)ý + 2yz 2).
Here k is a constant with the appropriate units. For the possible one, find the potential, using
the origin as your reference point. Check your answer by computing VV. [Hint: You must
select a specific path to integrate along. It doesn't matter what path you choose, since the
answer is path-independent, but you simply cannot integrate unless you have a particular path
in mind.]
An insulating solid sphere of radius 3 m has 15 C of charge uniformly distributed throughout its volume. Calculate the charge contained in a Gaussian surface having a radius 1/2 that of the sphere. Present your answer accurately to 2 decimal numbers i.e 3.20. Do not include units!
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- (Problem 4.10) A sphere of radius R carries a static radial polarization density P(r) = kr, r < R where k is a constant and r is the radial vector from the center of the sphere. (a) What are the dimensional units of the constant k? (b) Calculate the surface areal bound charge density o(R, 0, ø) and the volume bound charge density p(r). 2 (c) Find the electric field inside and outside the sphere.arrow_forwardA charge q = 54 C is located at (9, 8) and we want to find the electric field at pint p (3, 5). Find the source to point vector. Use the following constants if necessary. Coulomb constant, k = 8.987 × 10° N · m² /C² . Vacuum permitivity, €o = 8.854 × 10-12 F/m. Magnetic Permeability of vacuum, µo = 12.566370614356 ×x 10-7 H/m. Magnitude of the Charge of one electron, e = -1.60217662 × 10¬19 C. Mass of one 9.10938356 x 10-31 kg. Unless specified otherwise, each symbol carries their usual meaning. For example, µC means micro coulomb electron, me x component of the vector Give your answer up to at least three significance digits. y component of the vector Give your answer up to at least three significance digits.arrow_forwardConsider an infinitely long wire of charge carrying a positive charge density of A. The electric field due to λ this line of charge is given by E= 2kef= -, where is a unit vector directed radially outward Σπερμ from the infinitely long wire of charge. Hint #3 a. Letting the voltage be zero at some reference distance (V(ro) = 0), calculate the voltage due to this infinite line of charge at some distance r from the line of charge. Give your answer in terms of given quantities (A,ro,r) and physical constants (ke or Eo). Use underscore ("_") for subscripts and spell out Greek letters. Hint for V(r) calculation 3 V(r) = b. There is a reason we are not setting V(r → ∞o) = 0 as we normally do (in fact, in general, whenever you have an infinite charge distribution, this "universal reference" does not work; you need a localized charge distribution for this reference to work). Which of the following best describes what happens to potential as roo? (That is, what is V(ro), with our current…arrow_forward
- 1.27| The important dipole field (to be addressed in Chapter 4) is expressed in spherical coordinates as E =4 (2 cos 0 a, + sin 0 ag) where A is a constant, and where r> 0. See Figure 4.9 for a sketch. (a) Identify the surface on which the field is entirely perpendicular to the xy plane and express the field on that surface in cylindrical coordinates. (b) Identify the coordinate axis on which the field is entirely perpendicular to the xy plane and express the field there in cylindrical coordinates. (c) Specify the surface on which the field is entirely parallel to the xy plane.arrow_forwardAn Electron moves in the presence of a uniform magnesic field in The Z -direchin (B=Bk). Part a: write do wn a Vector potinh al A that generates the Specific magnhic field B. Assume A is independent of time and Az=0 Part b: Write down the darteoi ham iltonian for the system in terms of TTi , Where Tl; = P; -eAilc' is and e the electron Charge· * Part C:Compute [ TTe;T ly]arrow_forwardA charge located at the origin in free space produces a fields for which Ez=2kV/m at point P(-2, 4, -1). a. Find Q (in µC). b. Find E at D(3, 4, 2) in cartesian coordinates system. Kindly type/input the coefficients. c. Find E at D(3, 4, 2) in cylindrical coordinates system. Kindly type/input the coefficients. d. Find E at D(3, 4, 2) in spherical coordinates system. Kindly type/input the coefficients. Please show complete solution, thank you!arrow_forward
- Problem 1. Prove that for a vacuum-dielectric interface at glancing incidence ri→-1 (see Fig. 4.49 from textbook, also on slide 7 in Lecture 4). In the same figure, if a is the angle that the curve r(0.) makes with the vertical at 0; = 90°, then: Vn2 – 1 tana, 2 1.0 0.5 Op -0.5 56.3° -1.0 30 60 90 0; (degrees) Figure 4.49 The amplitude coefficients of reflection and transmission as a function of incident angle. These correspond to external reflection n; > n; at an air-glass interface (n = 1.5). Amplitude coefficients ofarrow_forwardAssume MKS units... Let Q be an open subset of R³. Let B: :Q - R³ be a continuous vector .field, representing a magnetic field in 3-D space. 7 Let P be a particle with charge q E R and mass m > 0. If p is at position (x. y, z) in Q and R³ is the velocity of p, at time t, then p feels a force 7(7,7) given by - 7(7,J) := q V × B (7) . Suppose that p moves along a curve C as time t varies from a to b, and that p has position vector (t) and instantaneous velocity (t) at time t. ř (1) Explain why the two vectors 7'(t) × È(7(t)) and 7'(t) are perpen- dicular at every time t = [a, b]. (2) Using Part (1), calculate W := the work done on the particle p by the force as p moves from D = 7(a) to E = √ (b) along C. F (3) Prove that ((t)||²)=27' (t) • F(t), at each time t. (4) Using Parts (2) and (3), and Newton's Second Law, prove that if the magnetic force - ₹(7,7) is the total force on p at every time t, then p moves along C at a constant speed. dtarrow_forwardProblem 8.4 (a) Consider two equal point charges q, separated by a distance 2a. Construct the plane equidistant from the two charges. By integrating Maxwell's stress tensor over this plane, determine the force of one charge on the other. (b) Do the same for charges that are opposite in sign.arrow_forward
- Problem 2.20 One of these is an impossible electrostatic field. Which one? (a) E= k[xy x + 2yz y + 3xz 2]; (b) E= k[y² + (2xy + z²)ŷ + 2yz2]. Here k is a constant with the appropriate units. For the possible one, find the poten- tial, using the origin as your reference point. Check your answer by computing VV. [Hint: You must select a specific path to integrate along. It doesn't matter what path you choose, since the answer is path-independent, but you simply cannot integrate unless you have a definite path in mind.]arrow_forwardUse the Divergence Theorem to find the outward flux of F = (9x° + 12xy) i+ (5y + 5e'sin z) j+ (9z° + 5e' cos z) k across the boundary of the region D: the solid region between the spheres x + y +z? = 1 and x2 +y? + z? = 2. ..... The outward flux of F = (9x + 12xy²) i+ (5y° + 5e'sin z) j+ (9z° + 5e'cos z) k is (Type an exact anwer, using n as needed.)arrow_forwardProblem 2.20 One of these is an impossible electrostatic field. Which one? (a) Ek[xy x + 2yzý + 3xz2]; (b) E= k[y² + (2xy + z²) ŷ + 2yz 2]. Here k is a constant with the appropriate units. For the possible one, find the potential, using the origin as your reference point. Check your answer by computing VV. [Hint: You must select a specific path to integrate along. It doesn't matter what path you choose, since the answer is path-independent, but you simply cannot integrate unless you have a particular path in mind.] Problem 2.11 Use Gauss's law to find the electric field inside and outside a spherical shell of radius R, which carries a uniform surface charge density o. Compare your answer to Prob. 2.7. Problem 2.21 Find the potential inside and outside a uniformly charged solid sphere whose radius is R and whose total charge is q. Use infinity as your reference point. Compute the gradient of V in each region, and check that it yields the correct field. Sketch V (r).arrow_forward
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