Concept explainers
- Beginning with the definition, of the scalar magnetic potential. H = ∇Ă—V, and using Eq. (25), show that in a region having permanent magnetization M and zero current a Poisson equation for magnetic potential can be developed:
where Pm = μ0 V. M is an equivalent magnetic charge density. What happens when M is uniform? (b) Using the definition of the
where Jeq = V x M in analogy to Eq. (57) in Chapter 7.
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Engineering Electromagnetics
- A charge QA = −20μC is located at A(−6, 4, 7), and a charge QB = 50μC is at B(5, 8,−2) in free space. If distances are given in meters, find:(a) RAB; (b) RAB. Determine the vector force exerted on QA by QB if 0 = (c) 10−9/(36π) F/m; (d) 8.854 × 10−12 F/m.Ans. 11ax + 4ay − 9az m; 14.76 m; 30.76ax + 11.184ay − 25.16az mN; 30.72ax+ 11.169ay − 25.13az mNarrow_forwardGiven the field: D = 20 [-sinØ ap + sin 2Ø aØ] Find the total charge lying within the volume 1<p < 2,0 < Ø < π, and 0 < z < 1 by evaluating both sides of the Divergence Theorem equationarrow_forwardThe image shows a coaxial cable (two nested cylinders) of length l, inner radius a and outer radius b. Note that l >> a and l >> b. The inner cylinder is charged to +Q and the outer cylinder is charged to −Q. a) Calculate the Poynting vector S in the cable.arrow_forward
- A charge QA = −20µC is located at A(−6, 4, 7), and a charge QB = 50µC is at B(5, 8, −2) in free space. If distances are given in meters, find: a. RAB; b. |RAB|. Determine the vector force exerted on QA by QB if E0 = c. 10−9/(36π) F/m; d. 8.854 × 10−12 F/m.arrow_forward2. If a plane charge of 12 µC/m2 lies on x = -6, determine electric flux density D at W(4, -9, -3). 3. If a line charge of 30 milliC/m is at y = -1 m and z = 4 m, find the vector force on a 40 pC charge at R(5,-15, 8). 4. Find the divergence of the vector: D = ρ2 cosΦaρ at (1, 30deg, -5)arrow_forwardConsider an infinitely large sheet lying in the xz-plane at the origin, the extent of the sheet in thex, and z direction is infinite. The sheet carries surface current J = J0ˆk, where J0 is the current per unit width perpendicular to the flow. Calculate the magnetic field everywhere (y > 0 and y < 0) due to J. Sketch the Amperian loop.arrow_forward
- Two very large concentric spherical shells of radii a=122m and b=389 m carry uniform charges -Q and Q as shown in the figure below. If the electric field between the shells is where Q=5μC, determine the potential difference between the shells; V(b)-V(a)=? Express your answer in units of Volts. Take the value of Coulomb constant as k=9x109 N.m2/C2. Hint: here the unit vector is radially outward.arrow_forwardA charge QA = 20μC is located at A(−3, 2, 4), QB = 30μC is located at B(2, 1, 4), and a charge QC = 50μC is at C(5, 8,−2) in free space. If distances are given in meters, find: (a) vector AB; (b) vector AC. Determine the total vector force exerted on QA by QB and QC. (Draw the diagram)arrow_forwardIn empty space (er = 1), the region x1 <x <x2 is filled with material with a dielectric coefficient er = 8 as a planar infinitely large plate. The geometry is shown in the figure below. Pay attention to the coordinate system placement.Region I: In (x <x1), the electric field vector was observed as E1 =(4.5) ux + (7.5) uy+(8.5) uz.Region II : (x1 <x <x2) de er = 8 by determining the electric field E2,Write the value of the x-component: E2x numerically.arrow_forward
- In free space, q1 = nCand LaTeX: q_2 = 4nC loads 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_forwardIn the upper half space, which is the empty space, I = 7 A current flows from the infinitely long wire along the y axis that intersects the z axis at the point C (0,0,10). Half-space z <0 is from a material with relative magnetic permeability µr = 5. Magnetic field in terms of given magnitudes Hx + Hy + Hz =? write it numerically.arrow_forwardLet D = 6xy3z2 ax + 7xyz4 ay + 8x5y4 az PC/m2 in free space, Answer all questions (a)Find the total electric flux passing through the surface y = 4, 0 ≤ x ≤ 3, 0 ≤ z ≤ 2 in a direction away from the origin. (b) Find the force |F| on the charge Q1(30 mC) at P(4,1,6)? (c) Find the total charge contained in an incremental sphere having a diameter of 8 mm centered at P (4, 1, 6)?arrow_forward
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