2.25 Consider the vector field rt r (r) = (p2 + E2)3/2 = (p2 + €2)372 (a) Calcul

Question

2.25 Consider the vector field rt r (r) = (p2 + E2)3/2 = (p2 + €2)372 (a) Calculate the divergence of F, and sketch a graph of the divergence as a function of r, for the three cases e 1. (b) Calculate the flux of F outward through a sphere of radius R centered on the origin, and verify it is equal to the integral of the divergence inside the sphere. (c) Show that in the limit E 0, the flux goes to 4 independent of R. Discussion: This example is important in electromagnetism: in the limit -0 , F has the form of the electric field of a point charge at rest at the origin. In this limit, the divergence of F approaches an infinitesimally narrow function of r with a finite volume integral of 4T.

Explanation / Answer

2.25

given vector field

F(r) = r/(r^2 + epsilon^2)^3/2

here r is a vector

a. let r = ai + yj + zk

r^2 = x^2 + y^2 + z^2

hence

divergence(F) = (d/dx i + d/dy j + d/dz k) x (Fxi + Fyj + Fzk)

div(F) = i(dFz/dy - dFy/dz) + j(dFx/dz - dFz/dx) + k(dFy/dx - dFx/dy)

now

Fx = x/(x^2 + y^2 + z^2 + e^2)^3/2

Fy = y/(x^2 + y^2 + z^2 + e^2)^3/2

Fz = z/(x^2 + y^2 + z^2 + e^2)^3/2

dFx/dy = -3xy/(x^2 + y^2 + z^2 + e^2)^5/2

dFx/dz = -3xz/(x^2 + y^2 + z^2 + e^2)^5/2

dFy/dx = -3yx/(x^2 + y^2 + z^2 + e^2)^5/2

dFy/dz = -3yz/(x^2 + y^2 + z^2 + e^2)^5/2

dFz/dx = -3zx/(x^2 + y^2 + z^2 + e^2)^5/2

dFz/dy = -3zy/(x^2 + y^2 + z^2 + e^2)^5/2

hence

div(F) = [i(-3zy +3yz) + j(-3xz + 3zx) + k(-3yx + 3yx)]/(x^2 + y^2 + z^2 + e^2)^5/2 = 0

b. flux outgoing at radius r is integral((F(r).dA))

now consider a ring on spohere at angle theta with xy plane

dA = 2*pi*r*cos(theta)*r*d(theta) = 2*pi*r^2cos(theta)d(theta)

at angle phi in xy plane

dA = r^2d(theta)d(phi)

now F(r) is directed in radial direction

hence dA.F(r) = flux

|F| = |r|/(r^2 + e^2)^3/2

flux = 2*pi*r^3*cos(theta)d(theta)/(r^2 + e^2)^3/2

integrating

flux = 2*pi*r^3(2)/(r^2 + e^2)^3/2 = 4*pi*r^3/(r^2 + e^2)^3/2

hence flux = integral(div(F)dV)

c. for e -> 0

flux = 4*pi

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