Adapted from Briggs, Cochran, and Gillett Calculus Within the study of electroma
ID: 2878522 • Letter: A
Question
Adapted from Briggs, Cochran, and Gillett Calculus Within the study of electromagnetism, charge density is a measure of electric charge per unit volume. The standard units for charge density are Coulombs per meters^3 abbreviated as C/m^3. Charge density may vary throughout a particular volume. A continuous function which describes a changing charge density is called a continuous charge distribution. Since charge density, D, is the ratio of electric charge to volume, we have that D = dQ/dV and therefore, the total charge, Q, within a volume is Q= tripleintegral DdV A spherical cloud of electric charge has a known charge density D(p), where p is the spherical coordinate. Find the total charge in the Interior of the cloud in the following cases. D(p) = 0.0002/p^4, 1 lessthanorequalto rho lessthanorequalto infinityExplanation / Answer
a) given D()=0.0002/4
in spherical coordinates
x=sincos,y=sinsin,z=cos
x2+y2+z2=2
dv=2sin d d d
0<=<=2,0<=<=,1<=<=
Q= [0 to 2] [0 to ] [1 to ] (0.0002/4)2sin d d d
Q= [0 to 2] [0 to ] [1 to ] (0.0002/2)sin d d d
Q= [0 to 2] [0 to ][1 to ] (-0.0002/)sin d d
Q= [0 to 2] [0 to ] ((-0.0002/)-(-0.0002/1))sin d d
Q= [0 to 2] [0 to ] (0+0.0002)sin d d
Q= [0 to 2][0 to ] 0.0002(-cos) d
Q= [0 to 2] 0.0002((-cos)-(-cos0)) d
Q= [0 to 2] 0.0002(2)
Q= 0.0002(2)(2-0)
Q= 0.0008
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b)
0<=<=2,0<=<=,0<=<=
Q= [0 to 2] [0 to ] [0 to ] (0.0002e0.013)2sin d d d
Q= [0 to 2] [0 to ][0 to ] (1/3)(-1/0.01)(0.0002e0.013)sin d d
Q= [0 to 2] [0 to ][0 to ] (1/3)(-0.02e0.013)sin d d
Q= [0 to 2] [0 to ] (1/3)((-0.02e)-(-0.02e0))sin d d
Q= [0 to 2] [0 to ] (1/3)(0+0.02)sin d d
Q= [0 to 2][0 to ] (0.02/3)(-cos)d
Q= [0 to 2] (0.02/3)((-cos)-(-cos0))d
Q= [0 to 2] (0.02/3)(1+1)d
Q= [0 to 2] (0.02/3)2
Q= (0.02/3)2(2-0)
Q= (0.08/3)
Q= 2/75
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