PHYS 3900 Homework Set #02

Transcription

1 PHYS 3900 Homework Set #02 Part = HWP 2.0, 2.02, Due: Mon. Jan. 22, 208, 4:00pm Part 2 = HWP 2.04, 2.05, Due: Fri. Jan. 26, 208, 4:00pm All textbook problems assigned, unless otherwise stated, are from the textbook by M. Boas Mathematical Methods in the Physical Sciences, 3rd ed. Textbook sections are identified as Ch.cc.ss for textbook Chapter cc, Section ss. Complete all HWPs assigned: only two of them will be graded; and you don t know which ones! Read all Hints before you proceed! Make use of the PHYS3900 Homework Toolbox, posted on the course web site. Do not use the calculator, unless so instructed! All arithmetic, to the extent required, is either elementary or given in the problem statement. State all your answers in terms of real-valued elementary functions (+,, /,, power, root,, exp, ln, sin, cos, tan, cot, arcsin, arccos, arctan, arcot,...) of integer numbers, e and π; in terms of i where needed; and in terms of specific input variables, as stated in each problem. So, for example, if the result is, say, ln(7/2) + (e 5 π/3), or 7 9/2, or (6 4π) 0, then just state that as your final answer: no need to evaluate it as decimal number by calculator! Simplify final results to the largest extent possible; e.g., reduce fractions of integers to the smallest denominator etc.. In some problems below, you will need to apply (without proof) d Alembert s Ratio Test (RT) which can be used to test convergence of a series of real or complex numbers S n := n T j, with n = 0,, 2,..., constructed from a sequence of real or complex numbers T j with j = 0,, 2,... The RT is applicable only if () an integer j o 0 exists such that T j 0 for all integer j j o ; and (2) the limit ρ := lim T j+ /T j j,j j o either exists with a finite limit ρ 0; or if it is is infinite (meaning: the sequence ρ j := T j+ /T j grows beyond all lower bounds for j ). The RT states: If ρ < the series S n is convergent, i.e., it does have a limit, S := T j ; and it is, in fact, absolutely convergent. If, on the other hand, ρ >, the series S n is divergent. If ρ = the series S n may be either convergent or divergent, i.e., the RT does not tell you anything about convergence. However, for power series, the condition ρ = often (but not always!) defines the boundary of the convergence disk: it can then be used to find both the radius, R c (a.k.a. the radius of convergence), and the center, z c, of the convergence disk.

2 HWP 02.0: Evaluate the following expressions, without any calculator, either in cartesian (CC) or polar (PC) coordinates, as stated. For CC, with z = x + iy, give x and y (with real x and y). For PC, with z = re iθ, give r and θ (with real r 0 and real θ). For PC, any θ with 0 θ 2π, but only those (Watch Out!), are acceptable. You may use, without proof, any tool in the Toolbox and any known property of real elementary functions. In parts (c) and (d), variables a, b, c and d denote real numbers. (a) ( 3i) 5 ( 2 + i 8) 7 (2 i) (CC); (b) ( 3 + i (CC); 2) (c) (7c 7ia) 3 (a 0 or c 0) (CC); 49i ( a + i c ) 3 (d) ( e i(b 2i ln(2)) e i(d+2i ln(2))) ( e 2iπ/3+id + e i(b 7π/3)), for cos(b + d)= 5 (e) i /i + i 3 3i 7 (PC). (PC); HWP 02.02: This problem extends the definition of the conventional, real square root function,..., to arbitrary complex numbers z = x + iy with x = Re z and y = Im z. It is a mathematically equivalent alternative to the definition of the square root of a complex number given in the Toolbox. (a) Show that, for every complex z = x + iy, there exists exactly one complex number w = u + iv, with u = Re w and v = Im w, obeying the following two conditions: and w 2 = z (2.2.) either Re w > 0 or [Re w = 0 and Im w 0] (2.2.2) Express both u and v as functions of x and y. This uniquely defined complex number w = u + iv is then referred to as the general complex square root of z, and denoted by c z := w. Hints: Write out Eq.(2.2.) as two real equations, expressed in terms of the real variables u, v, x and y. Eliminate v to solve for u; then solve for v. Prove that such a solution (u, v) always exists, for any choice of (x, y), while carefully distinguishing the two cases u 0 and u = 0. Prove that condition (2.2.2) makes the choice of this solution (u, v) unique, i.e., there is only one solution of Eq.(2.2.) which obeyes (2.2.2). (b) Use the results from Part (a) to calculate c +4, c 4, c 4i, c 4i, c + + i, c i, c + i, c + i. 2

3 Sketch each result as a point, properly labeled, in the complex plane, using your calculator to evaluate any real square roots you may need. (c) Use the result from Part (a) to prove that for purely real numbers, z=x with y=0, c z = x if x 0 ; c z = i x if x < 0 ; where... is the familiar, conventional square-root function defined for positive real numbers. (d) Consider any complex number z = x + iy written in polar coordinates, z = z e iθ, with θ chosen so that π < θ π. Use the result from Part (a) to show that c z = z e iθ/2. (e) Use the result from Part (d) to show for any non-zero complex z = x + iy 0 that c (/z) = /c z if x 0 or y 0 ; c (/z) = /c z if x < 0 and y = 0. HWP 02.03: Recall from your intro physics course that the electrical power dissipated in a circuit, i.e., the rate at which electrical energy is being consumed by the circuit, is the product of the voltage applied and the curent driven through the circuit by that voltage. This is a very general result, based only on the energy conservation law, and therefore holds even for time-dependent voltages and currents. (For an example, see the circuit shown in Fig below.) Therefore, if an alternating voltage Ṽ (t) := V o cos(ωt φ V ) drives an alternating current Ĩ(t) := I o cos(ωt φ I ) through an electrical circuit, the electrical power dissipated in the circuit, i.e. the circuit s rate of energy consumption at time t, is P (t) = Ṽ (t)ĩ(t). For practical purposes, knowing the exact t-dependence of this (rapidly) oscillating power function is often too much information : one is actually not interested in knowing the instantaneous power consumption P (t) at each time t, but rather only its long-time average over many periods oscillation, defined by: P = lim τ τ to+τ t o P (t)dt Note here, t o+τ t o dt P (t) is the amount of energy consumed in the circuit over a time interval [t o, t o + τ], of duration τ, starting at some time t o. (a) Show that P is independent of t o and that it can be written in terms of the complex voltage and current amplitudes V := V o e iφ V and I := I o e iφ I as P = 2 Re(V I). 3

4 While there are other stategies to get this result, in order to practice your complex calculus skills, you should solve this problem specifically in the following steps: () Express the real voltage and current as real parts of the complex voltage and current, by Ṽ (t) = Re(V eiωt ) and Ĩ(t) = Re(Ieiωt ). Use the identity Re(X) = (X + X )/2 (for any complex number X) to re-write the foregoing real parts in terms of V e iωt and Ie iωt and their respective complex conjugates, V e iωt and I e iωt, using (e iωt ) = e iωt. (2) Then take the product of Ṽ and Ĩ to express and expand out P (t) in terms of four products of V e iωt and Ie iωt and their respective complex conjugates, using also the identities e iωt e iωt = and e ±iωt e ±iωt = e ±2iωt. (3) For each of the resulting (four) V I-product terms in P (t), evaluate the definite t-integrals (from t o to t o + τ), by proving and then using the fact that for any complex z 0, F (t) := (iz) e izt is an indefinite integral (anti-derivative) of f(t) := e izt, i.e., df (t)/dt = f(t). You can prove this last equation, for example, by using the MacLaurin expansion of e izt in powers of t. (4) I trust you remember from your intro calculus course how to get from an indefinite to a definite integral! (5) Add up all the (four) definite integrals thus obtained, divide by τ and take the limit τ. Make use of the fact that e iθ = for any real θ, hence e iθ(τ) /τ 0 for τ and any real-valued function θ(τ). (6) Show that V I + V I = 2Re(V I) to complete the derivation. (b) Use the result from (a) and the definition of V and I, that is, V := V o e iφ V and I := I o e iφ I, to show that P can also be written as: P = 2 V oi o cos(φ) where φ is the phase difference between current and voltage: φ := φ I φ V. Footnote: Recall here that the t-dependent complex voltage V e iωt and t-dependent complex current Ie iωt represent rotating vectors in the complex plane, with the tips of the vectors moving along circles of radii V V o and I = I o, respectively, both of them rotating at angular velocity ω. The phase angle φ is then the constant angle enclosed between these two rotating vectors. Visualize this! It s an important way of thinking about sinusoidally oscillating quantities, that is useful far beyond just AC circuit analysis. (c) If the circuit is linear, i.e., if the current ampltitude is proportional to the voltage amplitude, then sinusoidally oscillating Ṽ (t) and Ĩ(t) are related by some complex-valued impedance function Z which is determined by the design ( innner workings ) of the circuit and generally dependent on angular oscillation frequency ω. That is, the corresponding complex amplitudes V and I then obey the generalized Ohm s Law: V = ZI. Let Z be written in polar coordinates as Z Z e iθ Z. Show that the result from (a) or (b) can then be written as V 2 o P = 2 Z I2 o cos(θ Z ) = 2 Z cos(θ Z). Hint: Show that the phase difference in (b) is φ = θ Z if V = ZI. 4

5 Fig c CtJ HWP 02.04: [Note: Before doing this problem, you may find it helpful to review the application of Kirchhoff rules to derive the equivalent resistance R for three ohmic resistors R, R 2 and R 3 connected in parallel to a battery in a simple DC circuit. The problem below, especially Part (a), is a generalization of these Kirchhoff rule ideas to an AC circuit.] An alternating current Ĩ(t) := I o cos(ωt φ I ) is driven by an applied voltage Ṽ (t) := V o cos(ωt φ V ) through the following parallel RLC-circuit, containing a resistor R with impedance Z R = R, an inductance L with impedance Z L = (iωl), and a capacitor C with impedance Z C = /(iωc), as shown in Fig (a) Show that the complex voltage and current amplitudes, V := V o e iφ V and I := I o e iφ I, are related by a generalized Ohm s Law, V = ZI, with an effective impedance Z (for the RLC-circuit as whole), given by: Z = + + = ( Z R Z L Z C R + i ωc ). ωl 5

6 Hints: Assume (without proof) the Kirchhoff junction rule stating for the case of this circuit that Ĩ(t) = ĨR(t) + ĨL(t) + ĨC(t) where ĨR(t) Re(I R e iωt ), ĨL(t) Re(I L e iωt ), and Ĩ C (t) Re(I C e iωt ) are the alternating currents flowing into R, L and C, respectively, as shown in Fig , with complex amplitudes I R, I L, and I C, respectively. Also assume (without proof) the Kirchhoff loop rule, stating for the case of this circuit that each of the three voltage drops, ṼR(t), ṼL(t), and ṼC(t), across the three circuit elements, R, L and C, respectively, is the same and equal to the applied voltage Ṽ (t). Re-state these Kirchhoff rule results in terms of the complex amplitudes I, I R, I L, I C, V, V R, V L, V C, associated with the corresponding real oscillating currents (Ĩ, ĨR, ĨL, ĨC) and voltages (Ṽ, ṼR, ṼL, ṼC). Then use (without proof) the fact that each of the three circuits elements R, L and C obeys it s own generalized Ohm s Law, namely, in terms of complex amplitudes: I R = Z R V R, I L = Z L V L, I C = Z C V C Use all the foregoing to first express I R, I C and I L in terms of V and impedances Z R, Z L and Z C ; and then I in terms of V and impedances Z R, Z L and Z C. Then simply use the definition Z := V/I to get the result for /Z stated above. (b) Use the results from Part (a) and from HWP Part (c) to show that the timeaveraged power dissipation P in the circuit, expressed as a function of I o, R, L, C and ω, can be written as: RI 2 o P = 2 + R 2 [ωc /(ωl)] 2 Sketch a very rough graph of P (ω) as a function of ω for fixed I o, R, L and C. Indicate on the graph the asymptotic behavior i.e., the approximate power laws P (ω) constant ω p for ω 0 and for ω. Hint: Let Y := /Z = Y e iθ Z. Find Re(Y ) and Im(Y ) and Y. Draw Y as a vector in the complex plane to prove/see that cos(φ) = Re(Y )/ Y with φ = θ Z. Also note that / Z = Y. Use Y instead of Z throughout the calculation. (c) Find the resonance frequency ω o, expressd in terms of R, L and C, where P (ω) reaches its maximum (at fixed I o ). Hint: Since P (ω) > 0, it reaches its maximum where /P (ω) has a minimum. Find the latter: it s easier! (d) Important: Calculate/prove all the following only at resonance, i.e., set ω = ω o. Calculate Z L and Z C and show that Z L = Z C. Use this to find P (ω o ) and phase angle φ = θ Z. Calculate the amplitudes I L,o I L and I C,o I C of the alternating currents ĨL(t) and Ĩ C (t), and show that I L,o = I C,o. 6

7 Prove that the L-current exactly cancels the C-current and that the total current equals the R-current, i.e., at resonance: Ĩ L (t) + ĨC(t) = 0 and Ĩ(t) = ĨR(t) at all times t Hints: First show Z L = Z C. Then, to calculate current amplitudes and show current cancellation, use the generalized Ohm s Law for complex amplitudes I, I L and I C to express I L and I C in terms of Y, Z L, Z C and I: I L = V/Z L = I/(Y Z L ), likewise I C = V/Z C = I/(Y Z C ). Hence, I L = I o /( Y Z L ) and I C = I o /( Y Z C ). Then use the results for Z L and Z C to show the cancellation in terms of the complex amplitudes: I L = I C. Hence, I L = I C and by Kirchoffs junction rule applied to the complex amplitudes, i.e., by I = I R + I L + I C, show that I = I R. Multiply the foregoing relations for the complex amplitudes by e iωot, take the real parts, and you ll get the corresponding desired relations for the real alterating currents, ĨL(t), ĨC(t), Ĩ(t) and ĨR(t). HWP 02.05: Let the MacLaurin series of the complex exponential function, the complex sine function, and the complex cosine function be defined, for any complex z and integer n 0, by exp n (z) := n j! zj, sin n (z) := n ( ) j (2j + )! z2j+, cos n (z) := n ( ) j (2j)! z2j Use d Alembert s Ratio Test to prove that each of these three MacLaurin series has an infintite radius of convergence, i.e., that each its limit exists for n and any complex number argument z, no matter how large z. Hints: To apply the RT to exp n (z), set T j = (/j!)z j ; to apply it to sin n (z), set T j = [( ) j /(2j + )!]z 2j+ ; etc.... Footnote: The corresponding infinite-series limits can then be used to define the complex generalizations of the exponential, sine and cosine functions. That is, for any complex argument z, one defines e z exp(z) := j! zj, sin(z) := ( ) j (2j + )! z2j+, cos(z) := ( ) j (2j)! z2j. For the special case that z x is a real number, the foregoing definition is consistent with, i.e., it conincides with, the elementary definitions of the real functions e x, sin x and cos x because of the fact that the latter three functions can be represented by exactly the same Mac Laurin series, with z being the real argument x. 7

8 HWP 02.06: Find the radius of convergence, R c, and the center of the convergence disk, z c, for each of the following four power series, using d Alembert s Ratio Test: m=0 (z i + 4) m 4m 5 + 6m 3 + 2, ( (j + 2) 9π 6 z ) 3j, 5 + 7i ( z ) j ( z ) j! ( ) j j!, ( ) j j!. i + 4 i + 4 8