Please help me find the electrical field generated by these gaussian spheres. Th

Question

Please help me find the electrical field generated by these gaussian spheres. Thanks!

Two spherical charge distributions sit on the x-axis so that they touch at the origin. Each sphere has radius R. Their volume charge densities depend on radius according to the formulas rho 1 = A/r and rho 2 = -B/r where A and B are positive contants with unit . With these charge distribution function, the charge inside Gaussian spheres of radius r drawn within the distribution are: q = 2 pi Ar 2 and q = 2 pi Br 2. Consider the electric field created by each sphere of charge and write the electric field vector for point P, Q, R, and S in terms of A, B, R, and epsilon 0. The exact coordinates are: P (-R, 0) Q (-R/2, ) R (0, 0) S (R, R) T (0, R)

Explanation / Answer

Note: e0 = epsilon0

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POINT "P":

Ex = q1/(4 pi e0 r1^2) + q2/(4 pi e0 r2^2)

Ex = (0)/(4 pi e0 R^2) + (2 pi B R^2)/(4 pi e0 (2R)^2)

Ex = (B)/(8 e0)

Ey = 0

Vector:

==> E(P) = ((B)/(8 e0) i^ + 0 j^

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POINT "Q":

Ex = q1/(4 pi e0 r1^2) + q2/(4 pi e0 r2^2)

q1 = (2 pi A R^2) * (pi (R/2)^2)/(pi R^2) = (2 pi A R^2) * (1/4) = (1/2) pi A R^2

Ex = ((1/2) pi A R^2)/(4 pi e0 (R/2)^2) + (2 pi B R^2)/(4 pi e0 (3R/2)^2)

Ex = ((1/2) pi A)/(4 pi e0 (1/2)^2) + (2 pi B)/(4 pi e0 (3/2)^2)

Ex = (A)/(2 e0) + (2 B)/(9 e0)

Ey = 0

==> E(Q) = (A)/(2 e0) + (2 B)/(9 e0) i^ + 0 j^

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POINT "R":

Ex = q1/(4 pi e0 r1^2) + q2/(4 pi e0 r2^2)

Ex = (2 pi A R^2)/(4 pi e0 R^2) + (2 pi B R^2)/(4 pi e0 R^2)

Ex = (A)/(2 e0) + (B)/(2 e0)

Ex = (A + B)/(2 e0)

Ey = 0

Vector:

==> E(R) = ((A + B)/(2 e0) i^ + 0 j^

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POINT "S":

Ex = q1/(4 pi e0 r1^2) cos(theta)

Ex = (2 pi A R^2)/(4 pi e0 (5 R^2)) * (2/(sqrt(5)))

Ex = (A)/(5 sqrt(5) e0)

Ey = q1/(4 pi e0 r1^2) sin(theta) - q2/(4 pi e0 r2^2)

Ey = (2 pi A R^2)/(4 pi e0 (5 R^2)) * (1/(sqrt(5))) - (2 pi B R^2)/(4 pi e0 (2R)^2)

Ey = (A)/(10 sqrt(5) e0) - (B)/(8 e0)

Vector:

==> E(S) = ((A)/(5 sqrt(5) e0) i^ + ((A)/(10 sqrt(5) e0) - (B)/(8 e0)) j^

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POINT "T":

Ex = q1/(4 pi e0 r1^2) cos(45) + q2/(4 pi e0 r2^2) cos(45)

Ex = (2 pi A R^2)/(4 pi e0 (sqrt(2) R)^2) (sqrt(2)/2) + (2 pi B R^2)/(4 pi e0 (sqrt(2) R)^2) (sqrt(2)/2)

Ex = (sqrt(2) A)/(8 e0) + (sqrt(2) B)/(8 e0)

Ex = (sqrt(2) (A + B))/(8 e0)

Ey = 0

Vector:

==> E(T) = (sqrt(2) (A + B))/(8 e0) i^ + 0 j^

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