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CHAPTER7 Solid angle, 3D integals, Gauss s Theoem, and a Delta Function We define the solid angle, subtended at a point P by a suface aea S, tobe ds() = S 3. (7.1) In this integal, ds is chosen on the side of S that typically makes ds positive, as shown in Figue 7.1. With this definition we can show that depends only on the position of P with espect to the peimete of S, because diffeent sufaces S and S that have the same peimete will subtend the same solid angle (povided P does not lie inside the closed suface fomed by S and S ). We ae claiming hee that ds() ds () = S 3 = S 3. (7.2) To pove this last esult, conside the closed suface fomed by S and S. Typically, one of the vectos ds and ds will point out of, and the othe will point into. As dawn in Figue 7.2, ds points in and ds points out. If the diection of d is defined eveywhee to point out of the closed suface, it follows that d S ds ds 3 = 3 S 3. But, by Gauss s theoem, ( ) d 3 = V 3 dv, whee V is the volume inside the closed suface (, ) and the point P lies outside V. It is easy to show (see Box 7.1) that 3 is zeo eveywhee inside V, hence d 3 = 0, and we can indeed use eithe S o S in the definition of. 123 woking pages fo Paul Richads class notes; do not copy o ciculate without pemission fom PGR 2004/11/3 10:50
124 Chapte 7 / SOLID ANGLE, 3D INTEGRALS, GAUSS S THEOREM, AND A DELTA FUNCTION FIGURE 7.1 This cudely illustates a 3D elationship between the point P and the suface S. In geneal neithe the suface no its peimete lies in a plane. is also equal to the aea on the unit sphee (i.e., a sphee with unit adius, centeed on P) cut by the staight lines fom P to the peimete of S. To see this, let S be the suface shown in Figue 7.3 and let A be the aea cut on the unit sphee. Then since = ds S 3 fom (7.2), and S is made up of aea A plus a suface fo which and ds ae pependicula, it follows that = da A 3. But we can simplify futhe, since is a unit vecto on A, and is nomal to da. Wefind = da= A. (7.3) A It is inteesting next to see what happens if instead of choosing two sufaces S and S as in Figue 7.2, we choose S and S as in Figue 7.4. The two sufaces still fom a closed suface, but now with P inside. We still take (the sum of two sufaces, hee S and S )as the closed suface, but now d, ds, and ds ae all outwad-pointing vectos. Recall fom (7.3) that = ds S 3 = A, whee A is the aea cut on the unit sphee. The total spheical aea is 4π,so ds S 3 = 4π A, and S + S = = 4π. But it is still tue that = ( ) V 3 dv, so why is this last integal not zeo, as in the pevious analysis when P was outside V? And why is it not zeo, given the esults developed in Box 7.1? ( ) The eason is that the poof of 3 = 0, given in Box 7.1, fails at the place whee = 0, i.e. at the point P itself. woking pages fo Paul Richads class notes; do not copy o ciculate without pemission fom PGR 2004/11/3 10:50
SOLID ANGLE, 3D INTEGRALS, GAUSS S THEOREM, AND A DELTA FUNCTION 125 FIGURE 7.2 Two sufaces, S and S, with the same peimete. ( ) We have now poved two vey impotant popeties of the expession 3. It is zeo eveywhee except at = 0 (as shown in Box 7.1). And the volume integal ( ) V 3 dv is eithe zeo o 4π, depending on whethe the point P lies outside o inside the volume. Theefoe the integand must be a delta function. In fact, it must be 4π times the standad Diac delta function in thee-dimensional space. The mathematical units in which a solid angle is measued ae called steadians (compae with adians, the mathematical units fo odinay angle). The solid angle epesenting the totality of all diections away fom a point is 4π steadians (compae with the value 2π adians fo the odinay angle coesponding to all diections away fom a point in a plane). The steadian is physically a dimensionless unit (just as adians and degees ae dimensionless). Finally, note that of solid angles that ( ) ( ) 1 3 = 2. We have theefoe shown by ou discussion 2 ( 1 ) = 4πδ(), woking pages fo Paul Richads class notes; do not copy o ciculate without pemission fom PGR 2004/11/3 10:50
126 Chapte 7 / SOLID ANGLE, 3D INTEGRALS, GAUSS S THEOREM, AND A DELTA FUNCTION BOX 7.1 Two popeties of a paticula vecto elated to gavitation At the heat of ou discussion of solid angle is the vecto 3. Fist we shall evaluate the divegence of this vecto, and then we ll elate the vecto to the gadient of a scala. Povided 0, we can wite ( ) 3 = ( ) xi 3 = 3 3 3 x i 4. (1) But 2 = x j x j, and diffeentiating this esult with espect to x i we obtain 2 = x j 2x j = 2x j δ ij = 2x i, and hence = x i. (2) It follows fom (1) and (2) that ( ) 3 = 3 3 3 x i x i 4 = 3 2 3 3 = 0. (3) 5 Next, we can note that the vecto 3 is elated to the gadient of the scala 1. This follows because, using components in a catesian system, ( ) 1 = ( ) 1 = 1 2 = x i 3 = 3, i i and so 3 = ( ) 1. (4) unit vecto in the diection of inceasing Note that = 3 2. Theefoe, in the context of gavity theoy, we can ecognize the vecto as having magnitude and diection like 3 those of an invese squae law fo an attactive foce between two paticles a distance apat. Equation (4) gives the associated potential. Putting the esults (3) and (4) togethe, we find that ( ) 1 2 = 0 povided 0. (5) woking pages fo Paul Richads class notes; do not copy o ciculate without pemission fom PGR 2004/11/3 10:50
Poblems 127 aea A FIGURE 7.3 Solid angle is shown as an aea A pojected fom S onto pat of the unit sphee. The aea S is made up fom aea A plus the pat of a cone between the peimete of A and the peimete of S. whee δ() is the thee-dimensional Diac delta function. Moe geneally, ( ) 1 2 = 4πδ(x ξ). x ξ Poblems 7.1 What is the solid angle subtended by the blackboad in Schemehon Room 555 in the following cases: (a) at the eye of a peson in the middle of the main ow whee people sit; (b) at the eye of a peson in a fa cone of the oom; and (c) at the eye of a myopic (shot-sighted) instucto with his o he nose ight up against the boad? (Give appoximate answes, in steadians.) woking pages fo Paul Richads class notes; do not copy o ciculate without pemission fom PGR 2004/11/3 10:50
128 Chapte 7 / SOLID ANGLE, 3D INTEGRALS, GAUSS S THEOREM, AND A DELTA FUNCTION suface S suface S P FIGURE 7.4 P is now inside the closed suface fomed by S and S (which shae a common peimete). woking pages fo Paul Richads class notes; do not copy o ciculate without pemission fom PGR 2004/11/3 10:50