Engineering Electromagnetics
9th Edition
ISBN: 9780078028151
Author: Hayt, William H. (william Hart), Jr, BUCK, John A.
Publisher: Mcgraw-hill Education,
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Chapter 4, Problem 4.25P
Consider an electric field intensity in free space that exhibits a Gaussian function of radius in spherical coordinates
where pc and a are constants, and where the field exists everywhere, (a) What charge density would produce this field? (b) What total charge is present? (c) What is the potential at the origin? (d) What is the net stored energy in the field?
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Volume charge density is located in free space as ρν = 2e−100r nC/m3 for 0 < r < 3 mm, and ρν = 0 elsewhere.
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In free space, there are 2 concentric spherical surfaces with radii a and b. There is a constant volume charge density between 2 surfaces such that: ρ= { 0; ρb [ ? ?3 ] a) Find the electric potential everywhere using Poisson’s equation. b) Find the electric field vector everywhere using Poisson’s equation
A charge located at the origin in free space produces a fields for which Ez=2kV/m at point P(-2, 4, -1)
Find E at D(3, 4, 2) in spherical coordinates system.
Chapter 4 Solutions
Engineering Electromagnetics
Ch. 4 - Given E = Exax + Eyay + Ez3z V/m, where EX, Ey,...Ch. 4 - A positive point charge of magnitude q1 lies at...Ch. 4 - Given E=Epap+Ea+Ez+azV/m, where Ep, E and E2 are...Ch. 4 - An electric field in free space is given by...Ch. 4 - Consider the vector field G = (A/p) aa where A is...Ch. 4 - A electric field in free space is given as...Ch. 4 - Prob. 4.7PCh. 4 - Given E=-xax+yay,(a) find the work involved in...Ch. 4 - An electric field intensity in spherical...Ch. 4 - A sphere of radios a carries a surface density of...
Ch. 4 - At large distances from a dipole antenna (to be...Ch. 4 - Prob. 4.12PCh. 4 - Thee identical point charges of 4 pC each are...Ch. 4 - Given the electric field E=(y+1)ax+(x1)ay+2az find...Ch. 4 - Two uniform lines, 8 nC/m, are located at x=1, z=2...Ch. 4 - A spherically symmetric charge distribution in...Ch. 4 - Uniform surface charge densities of 6 and 2 nC/m2...Ch. 4 - Find the potential at the origin produced by a...Ch. 4 - Volume charge density is given as pv=poer/C/m3,...Ch. 4 - En a certain medium, the electric potential is...Ch. 4 - Prob. 4.21PCh. 4 - A Line charge of infinite length lies along the z...Ch. 4 - Prob. 4.23PCh. 4 - A certain spherically symmetric charge...Ch. 4 - Consider an electric field intensity in free space...Ch. 4 - Let us assume that we have a very thin, square,...Ch. 4 - By performing an appropriate Line integral from...Ch. 4 - Prob. 4.28PCh. 4 - A dipole having a moment P=3ax-5ay+10aznC.m is...Ch. 4 - Prob. 4.30PCh. 4 - A potential field in free space is expressed as...Ch. 4 - Prob. 4.32PCh. 4 - Prob. 4.33PCh. 4 - A sphere of radius a contains volume charge of...Ch. 4 - Four 0.8 nC point charge are located in free space...Ch. 4 - Surface charge of uniform density ps lies on a...
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- A sphere of radius R has total charge Q. The volume charge density (C/m3) within the sphere is ρ(r)=C/r2, where C is a constant to be determined. 1. Use Gauss's law to find an expression for the magnitude of the electric field strength E inside the sphere, r≤R, and r>- (more or equal) in terms of Q and R.arrow_forwardTwo spherical charges with radius R = 1m and radius R = 1.5m in free space appear as. The volumetric charge densities in both spheres are the same and ρv1 = ρv2 = 3μC / m3. The distance between the two spheres from outside to outside is d = 0.5m. Find the magnitude and direction of the electric field at point P on the circle of radius R = 1m from the center of the large sphere?arrow_forwardGiven a 60-uC point charge located at the origin, find the total electric flux passing through: (a) that portion of the spherer= 26 cm bounded by 0 < 0 < and = 0 <0< (b) the closed surface defined by p = 26 cm and := ±26 cm; (c) the plane := 26 cm.arrow_forward
- A uniform line charge and a uniform sheet charge, both infinite in extent, are located in free space along the y-axis and at x = -1 respectively. Determine E at point M(2, -1, 1) if the line charge density is 15 nc/m and the sheet charge density is 2 nC/m^2arrow_forwardA solid conducting sphere carrying charge q has radius a. It is inside a concentric hollow conducting sphere with inner radius b and outer radius c. The hollow sphere has no net charge. (a) What is the charge on the inner surface of the hollow sphere? On the outer surface? (b) Sketch the electric field lines of the system. (c) Derive expressions for the electric field magnitude and for the electric potential interms of the distance r from the center for the regions r < a, a < r < b, b < r < c, and r > c. (Take the electric potential to vanish at r = ∞.)(d) Graph the magnitude of the electric field and the electric potential as a function of r from r = 0 to r = 2c.arrow_forwardA charge located at the origin in free space produces a fields for which Ez=2kV/m at point P(-2, 4, -1). Find Q in microcoulombs.arrow_forward
- In free space, q1 = 3nC and q2 = 4nC charges are placed at y = 1 and y = 6, respectively, on the y-axis as shown in the figure. Accordingly, what is the electric potential value at y = 4?arrow_forwardA sphere with a radius of 1 relative dielectric constant 0 = 10 has a volumetric charge distribution varying only by "r" (independent of θ and Φ). (Here r is the distance measured from its origin in spherical coordinates.). The region outside the sphere that opens to infinity is the void. The electric field vector in the region (r <1) inside the sphere is as in the photo. Find the explicit expression of the electric field vector outside the sphere.arrow_forwardTwo point charges with ?1 = 2 × 10 ^−5 ? and ?2 = −4 × 10 ^−5 ? in Cartesian coordinates in free spaceIt is located at (1,3, −1) and (−3,1, −2) points, respectively. (a) Electricity at point (3,1, −3)(b) Find the force acting on an 8 × 10 ^−5 ? charge located at its origin.arrow_forward
- A 5-nC póint charge is located at A(2, -1, -3) in free space. (a) Find E at the origin.(b) Plot E(x, 0,0)| vs. x, -10<x<10 m.(c) What is |E(x,0,0)|max.?arrow_forwardA finite line charge located at (0, y, 0)m, where a≤y≤b , has a line charge density of ρL = ycosy C/m. Determine the integral that will solve for the electric field at the origin.arrow_forwardCalculate the electric field intensity E at M(3, -4, 2) in free space caused by: a. Charge Q1 = 2 µC at P1(0, 0, 0) b. Charge Q2 = 3 µC at P2(-1, 2, 3) c. Charge Q3 = 4 µC at P3(-3, -2, 4) d. The total field intensity of the 3 charges ( please provide complete solutions, thank you )arrow_forward
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