A sphere of radius R has total charge Q. The volume charge density (C/m3) within the sphere is rho(r)=C/r2, where C is a constant to be dete

Question

A sphere of radius R has total charge Q. The volume charge density (C/m3) within the sphere is rho(r)=C/r2, where C is a constant to be determined. Part A The charge within a small volume dV is dq=rhodV. The integral of rhodV over the entire volume of the sphere is the total charge Q. Use this fact to determine the constant C in terms of Q and R. Hint: Let dV be a spherical shell of radius r and thickness dr. What is the volume of such a shell? Express your answer in terms of the variables Q and R.

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Thiên Thanh 5 years 2021-07-14T14:07:48+00:00 1 Answers 74 views 0

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  1. thanhcong
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    2021-07-14T14:09:35+00:00
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    Answer: C = Q/4πR

    Explanation:

    Volume(V) of a sphere = 4πr^3

    Charge within a small volume ‘dV’ is given by:

    dq = ρ(r)dV

    ρ(r) = C/r^2

    Volume(V) of a sphere = 4/3(πr^3)

    dV/dr = (4/3)×3πr^2

    dV = 4πr^2dr

    Therefore,

    dq = ρ(r)dV ; dq =ρ(r)4πr^2dr

    dq = C/r^2[4πr^2dr]

    dq = 4Cπdr

    FOR TOTAL CHANGE ‘Q’, we integrate dq

    ∫dq = ∫4Cπdr at r = R and r = 0

    ∫4Cπdr = 4Cπr

    Q = 4Cπ(R – 0)

    Q = 4CπR – 0

    Q = 4CπR

    C = Q/4πR

    The value of C in terms of Q and R is [Q/4πR]

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