agn constant ,, to half-fill a parallel-plate capacitor (Fig. 4.25). By what fraction the capacitance increased when you distribute the material as in Fig. 4.25(a)? Hc about Fig. 4.25(b)? For a given potential difference V between the plates, find D, and P, in each region, and the free and bound charge on all surfaces, for ba cases.
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- § 2.10 Clampers Sketch v, for each network of Fig. 2.161 for the input shown. Would it be a good approxima- tion to consider the diode to be ideal for both configurations? Why? 120 V Si -20 V (a) (b) Figure 2.161I solved it but I need help in two parts.For first part, How to show thev formula is a solution of ODE? Second, for the third part, how to show it is bounded because I can not integratw matrix?The circuit shown in Figure 4.55 contains two nonlinear devices and a current source. The characteristics of the two devices are given. Determine the voltage, v, for (a) is = 1 A, (b) is = 10 A, (c) is 10 A, (c) is = 1 cos (t). is N₁ N2 i2 + i₁(A) ν -1 v (V) i2(A) + v (V)
- Figure 1.52 shows a spherical shell of charge, of radiusa and surface density σ, from which a small circular piece of radius b << ahas been removed. What is the direction and magnitude of the fieldat the midpoint of the aperture? Solve this exercise using the relationship for a force on a small patch.2.5.2 (a) (b) From the results of Exercise 2.5.1, calculate the partial derivatives of f. 6, and with respect to r, e, and . With V given by 1a rsine ap är (greatest space rate of change), use the results of part (a) to calculate V-Vy. This is an alternate derivation of the Laplacian. Note. The derivatives of the left-hand V operate on the unit vectors of the right-hand V before the unit vectors are dotted together.Figure 1.52 shows a spherical shell of charge, of radiusa and surface density σ, from which a small circular piece of radius b << a has been removed. What is the direction and magnitude of the field at the midpoint of the aperture? Solve this exercise using direct integration.
- What is the magntiude of the integral of B over dl for part c? Enter in your answer in micro-T*m. Assume that all currents are 11.1 A.Problem 4.45 What is the probability that an electron in the ground state of hydro- gen will be found inside the nucleus? (a) First calculate the exact answer, assuming the wave function (Equation 4.80) is correct all the way down to r = 0. Let b be the radius of the nucleus. (b) Expand your result as a power series in the small number € = 2b/a, and show that the lowest-order term is the cubic: P≈ (4/3)(b/a)³. This should be a suitable approximation, provided that bProblem 7.63 Prove Alfven's theorem: In a perfectly conducting fluid (say, a gas of free electrons), the magnetic flux through any closed loop moving with the fluid is constant in time. (The magnetic field lines are, as it were, "frozen" into the fluid.) (a) Use Ohm's law, in the form of Eq. 7.2, together with Faraday's law, to prove that if o ∞o and J is finite, then 7.3 Maxwell's Equations a slightly different approach to the same problem, see W. K. Terry, Am. J. Phys. 50, 742 (1982). R ƏB at S do= =V x (v x B). FIGURE 7.58 (b) Let S be the surface bounded by the loop (P) at time t, and S' a surface bounded by the loop in its new position (P) at time t + dt (see Fig. 7.58). The change in flux is S' dt B(t + dt) da + do = dt S B(t + dt) da - Use VB0 to show that S √₂³ (where R is the "ribbon" joining P and P'), and hence that ав l vdt at dt B(t + dt) da = S B(t). da. da - R S B(t + dt) da B(t + dt) da (for infinitesimal dt). Use the method of Sect. 7.1.3 to rewrite the second inte-…