Question

Q1. DELTA FUNCTIONS! a) Calculate the following integrals, assuming c=2 in all parts! In part iv, assume the volume V is a sphere of radius 1 centered on the origin, and the constant vector ro = (0,3,4) c 8(x-c)dx (Hint: watch out for limits of integration. Remember, c=2 throughout) 8(x-c)dx iii) Sºx-cl 8(2x)dx (Hint: see Griffiths Example 1.15) iv) SS ir-r, 8(r)dr v) M F-F, P 8°(27)di (Hint: 8' () = 8(x)O(y)8(z)) b) Evaluate the integral ſ (1+e") ( )dt where volume V is a sphere of radius R centered at the origin, by two different methods, just as Griffiths does in his Ex 1.16. (Hints: one method, surely the easier one, is to use your knowledge of what V is (Griffiths section 1.5.3 covers this) The other is to use "integration by parts in 3-D", described in Griffiths section 1.3.6. Don't forget that the inside flyleaf of Griffiths tells you how to take div, grad, and curl in, e.g. spherical coordinates!

Answer 1

Page N (Date: (4 1201 Dirac delta function is defined as Zoo te 8(2-a) dx = 1 . fo 8(x-2) dx (62) i Jo 8(e-c) dx using the property f(x) 8(x-a) dx c geven f(a) if aca<B if a< A & a> B 2 J 8(x-2) dx .0 .: 271} ) Rox 8(x-c) dx = 10 10 -lo 4x S(x-2) dx = 8 = 4 x 8(x-2) dan Ho ( 2) 4(2) s using I property w - where & f(x) = 4x here y hows en = (x-21² 8 (22) dx J 1x-cl² S(2x) dx -00 J (x-2 1² 8 (28) dx = ? is using the property s[kx) - 860) (x-21² , 8(x) dx = 1/(X-2 1² & (x-0) dx - 1 10-21" - 2 l: of property ) (2)

(V) IS A tal? 8° (d In 3-dimensions, property can be more fllflor) 8 (M-a) dt = f(2) alised as SI À 8°(H) di 10 -ño 1? kawa ở - P = (2 - 551 + (O-3) + ( 4 ) { - T02+(-3)* + (434125 SIJ 14 - lepe sa dê .5.- Ⓡ ♡ I TH Fotº 83 (28) de a In 3-D dimensions go (ků) = f sch) using this property Lū - Do 8°(2%) di H = 1-2012 1.8°(8)di = HA-A013182X H 10 = I x 5 wing of result

B) 1 (1+e*) (v. À Jdt = ? uthods we know the most general formula ' = 483() stretty 88 417 83 ) dt - futet )V.). >/([tex). Vill dt. 41 ((1 teady 8°(2%) at 842 / using property ③ Method 2 UTT X2 = 8 for Integuating above ear by parts, we. require the following Let us consider pr. (fa) dł Now using divergence theorem & dveragina . (fQ) dt a f( 7 ) d t + / 2. (7f) dt & 17. (f 8) dt = & F, a dig - (d) from (C) & (d) egy f lū. a) dt + [@lýt)dt = $ FQ. ds & 1 f (7.6) dt = -200 (7 f) dr + If o dř Here we have a = ) - a(e) re & f = lltex) de

д HP i using the result (e), we get (lte") V. Adt :-( . 8 (1+ & M) d t + ( +"); & 7 (lt e") = 2 (1+eM) - e-te lete*) . de -- (-e-4) (unité det). + (1+e") i to in o do do i - +90 let ar + 104e* */ sino doaap Ó I PER = 470 etto + (1+e+R) 417 --47 le *- 1) + 401 + 40 € * -unlo-1) + UT +0 4T + UT ( HueRoo, 8T