Problem 4.12 Calculate the potential of a uniformly polarized sphere (Ex. 4.2) directly from Eq. 4.9. 1 P(r). V (r) = = dt'. (4.9) Απερ 22 V
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- Example 7.7. A uniform magnetic field B(1), pointing straight up, fills the shaded circular region of Fig. 7.25. If B is changing with time, what is the in- duced electric field? Solution E points in the circumferential direction, just like the magnetic field inside a long straight wire carrying a uniform current density. Draw an Amperian loop of radius s, and apply Faraday's law: do E-d1 = E(27s) = (s²B(1)) =-7S 2 dB di dt Therefore s dB E 2 dr If B is increasing, E runs clockwise, as viewed from above. 13 Magnetostatics holds only for time-independent currents, but there is no such restriction on 3B/ar. Chapter 7 Electrodynamics B(1) E Amperian loop FIGURE 7.25 dt Rotation direction dl λ FIGURE 7.26How would I do part A.9.3. A system of two conductors has a cross section given by the intersection of two circles of radius b with centers separated by 2a as shown in figure below. The conducting portion is shown hatched, the non-hatched lens-shaped region being a vacuum. The conductor on the left carries a uniform current density, J1, coming out of the page, and the conductor on the right carries a uniform current density, −J 12, going into the page. Assume that the magnetic permeability of the conductor is the same as that of the vacuum. Find the magnetic field at all points in the vacuum enclosed between the two conductors. 2a
- PROBLEMS Section 6.2-Poisson's and Laplace's Equations 6.1 Given V -5x'y'z and e = 2.25e find (a) E at point P(-3, 1, 2), (b) p, at P. Conducting chete ore le and y 3. planes The space between them isProblem 9.4 Consider an infinitely long straight string whose linear mass den- sity is A (mass per unit length). By direct integration, determine the gravita- tional field a distance r from the string.4) a) What does Helmholtz theorem say? One sentence, plus 3 equations. b) Apply Helmholtz to some generic current density 7(7). Then assume that you have the magnetostatic case. Simplify. Then, write the vector potential as a function of B. Use curl B=.. c) Prove that div Ã(7) = 0 in the "minimal gauge". d) Derive Á(7) as a function of B (†) in the "minimal gauge" in the magnetostatic case.
- Problem 5.7 For a configuration of charges and currents confined within a volume V, show that LJ Jdr = dp/dt, (5.31) where p is the total dipole moment. [Hint: evaluate V (xJ) dt.] BNeed help with this problemProblem #3: Problem #3: FindS Enter your answer symbolically, as in these examples Use the divergence theorem to find the outward flux SS, vector field F = cos(9y+9z)i + 2 ln(x² + 7z)j + 3z² k, where S is the surface of the region bounded within by the graphs of z = √16-x²-1², x² + y² = 6, and z = 0. of the
- Problem 4.15 A thick spherical shell (inner radius a, outer radius b) is made of dielectric material with a "frozen-in" polarization P(r) = i where k is a constant and r is the distance from the center (Fig. 4.18). (There is no free charge in the problem.) Find the electric field in all three regions by two different methods: AP b P P (a) Sphere (b) Needle (c) Wafer FIGURE 4.19 FIGURE 4.18Problem 10.7A time-dependent point charge q (t) at the origin, p (r, t) = q (t)8³ (r), is fed by a current J(r, t) = − (1/4)(ġ/r²) î, where q = dq/dt. (a) Check that charge is conserved, by confirming that the continuity equation is obeyed. (b) Find the scalar and vector potentials in the Coulomb gauge. If you get stuck, try working on (c) first. (c) Find the fields, and check that they satisfy all of Maxwell's equations.³Please don't provide handwritten solution ....