This is a classical Physics Homework it has 12 problems most of them deal with electrical charge
Microsoft Word - Homework 3.docx 1 Applied Physics 615.411: Classical Physics B.G. Boone, Instructor April 23, 2018 Homework #3 (due end of semester) 1. What is the force on a point charge of Q1 = 50 µC located at (0,0,5) m on the z-axis due to a charge of Q2 = 500p mC, which is uniformly distributed over a circular disk of radius r = 5 m at z = 0 m. Determine charge density rs = Q2/A, (where Q2 is the total integrated charge on the disk). Define S in cylindrical coordinates using unit vectors ?"#and ?"%. Construct the differential force in terms of the product of the charge Q1, and the differential charge from a small sector segment as shown below. ?'(((⃑ = ?'?, 4??0?,', ?",' where: ?(⃑ ,' = ?,'?",' 2. What net flux crosses the closed surface S shown below, which contains a charge distribution in the form of a plane disk of radius 4 m with a density rs = sin2f/2r [C/m2]? Note, in this case the net flux corresponds to the charge because in the absence of a negative charge, the flux from the source terminates at infinity. So we must calculate: Φ = 3 3 ?5 6 7 ,8 7 ????? 3. Find the work done in moving charge Q = 5 mC from the origin to (2m, p/4, p/2) in spherical coordinates in an electric field given by: ?(⃑ = 5?? # 6?"# + 10 ????? ? "G [? ?⁄ ] 2 The differential work is dW = -QE× dl. We choose a path given below. Along path I, dq = df = 0 so only the radial component matters. Write down that result. Along path II dr = dq = 0. Write down the result again. Then, along path III dr = df = 0 . Write -down that result. Then, combine all tree results and integrate them over the residual integrals to get the actual work done [J]. 4. The continuity equation: ∇((⃑ ∙ ? = −QR QS can be rewritten using J = sE = (s/e)D as: ∇((⃑ ∙ ? ? ? ((⃑ = − ?? ?? If s and e are constant, then: X Y Ñ((⃑ ∙ ?((⃑ = X Y ? = − QR QS Solve for the time dependence of r(t). Given s = 6.17 x 107 [Siemens/m] for silver, where the charge distribution density, ro is in a block of silver. Find r after 1 and 5 time constants. First determine t = e/s , assuming e = eo. 5. Determine the resistance of the insulation in a length, l of coaxial cable as shown below given a current I between the inner and outer conductors. At a radial distance r determine the current density J in terms of the geometry. Then, determine the electric field and from that the voltage difference. Then determine the resistance. 6. Given an electric dipole oriented along the z-axis consisting of a point charge +Q at z = d/2 and a point charge -Q at z = -d/2, as shown below, the dipole moment is p = Q/d. Write- down the potential at point P. A dipole at the origin is obtained in the limit as d ® 0. For small d: r2 – r1 » dcosq2 » dcosq and r1r2 » r2. Then, determine the potential in the limit. 3 7. Consider the vector potential, A in the region surrounding an infinitely long, straight filamentary current I. The current element is: Idl = I dl?"%. The vector potential is given by: ? = 3 ?0??? 4?? ^ ?^ ?"% For increasing l, R® l, so in the limit of l ® ¥, A does not exist (verify by inspection). However, we can consider the differential vector potential given by: ?? = ?0??? 4?? ? "% Then, consider just the current density element at the origin (where l = 0). Rewrite dA in Cartesian coordinates. Then use the expression dB = Ñ× dA. Does this agree with the Biot- Savart law? 8. In a region surrounding the origin B = 5 x 10-4 ?"% [T] and E = 5 ?"% [V/m]. A proton with charge Qp = 1.602 x 10-19 C and mass mp = 1.673 x 10-27 kg enters this region at the origin with an initial velocity ?0= 2.5 x 105 ?"` [m/s]. Describe the proton’s motion after three complete revolutions. Use the Lorentz force equation: �⃑�a = ?ab?(⃑ + ?0 × ?(⃑ e. Consider first only the electric field to determine z(t). Then, consider the magnetic component. What type of motion does it induce? What is the characteristic period of this motion? Describe the combined motion and sketch and label your diagram. After three revolutions (where x = y = 0) determine the value of z. 9. A circular disk as shown rotates at w [rad/s] in a uniform magnetic induction of B = B?"%. Sliding contacts connect a voltmeter to the disk. What voltage is indicated on the meter from this arrangement? (This is the Faraday homopolar generator.) Consider one radial element. In general, the radial element has velocity v = w r ?"G . Use the Lorentz force law to determine the E-field. Then, solve for the voltage from: ? = 3 ?(⃑ f 7 ∙ ?? 4 10. Given the field E(z,t) = 10sin(wt + bz) ?"`+ 10cos(wt + bz) ?"g in the z = 0 plane, determine the actual values for the Ex and Ey components for wt = 0, p/4, p/2, 3p/4 and p. 11. In free space E(z,t) = 50cos(wt -‐ bz)?"` [V/m]. Find the average power crossing a circular area of radius = 2.5 m in the plane of z = constant. Write E in complex form. The impedance of free space is h = 120p W. Find H [A/m] and then calculate: Pave = ½ Re[E x H*] Reduce this to the number of Watts. 12. For a Hertzian dipole shown below, the vector potential is given by ?(?) = ???kl# 4?? (???)?"% where in spherical coordinates: ?"% = ?????"# − ?????"o Using H = Ñ x A/µ and E =Ñ x H/jwe, find the relevant field components, defined as Hf, Er and Eq.