Flux and nonconducting shells. A charged particle is suspended at the center of
ID: 1999916 • Letter: F
Question
Flux and nonconducting shells. A charged particle is suspended at the center of two concentric spherical shells that are very thin and made of nonconducting material. Figure (a) shows a cross section. Figure (b) gives the net flux ? through a Gaussian sphere centered on the particle, as a function of the radius r of the sphere. The scale of the vertical axis is set by ?s = 4.0 × 105 N·m2/C. (a) What is the charge of the central particle? What are the net charges of (b) shell A and (c) shell B?
4 (105 N m2/C)Explanation / Answer
the region 0<r<4 inside sphere A;
Gauss says: =ds*E(,,r), WHERE
ds*E(,,r) is dot product of 2 vectors ds and E(,,r),
|ds|=r*sin*d*r*d is elementary area on a sphere with radius r in spherical coordinate system, direction of ds being normal to the sphere,
|E(,,r)|=-q/(4*0*r^2) is strength of electric field produced by a point charge q in the center, direction of E being normal to the sphere, 0=8.854 x 10^-12 is const,
angle is measured around z-axis as 0<=<2,
angle is measured from plane XOY as –/2<=<= /2;
Since vectors ds and E(,,r) are parallel then
ds*E(,,r) = |ds|*|E(,,r)|;
= - r*sin*d*r*d * q/(4*0*r^2)
= -q/(4*0) d sin*d
=q/(4*0) d cos {=-/2 to /2}
= q/(4*0) d (1+1) {=0 to 2}
= 4*q/(4*0) = q/0 =2e5 N*m^2/C according to your graph;
hence q=8.854 x 10^-12 x 2 x 10^5 =1.77C
Gauss says in general: =q[k], where
k=1 to n, n is number of charges q[k] inside of a closed surface s;
applying
we get q(A) =8.854 x 10-12 x 4 x 10^5 = -3.54 C
we get q(B) =8.854 x 10^-12 x 6 x 10^5 =5.31 C