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Find the capacitance of an isolated ball...

Find the capacitance of an isolated ball-shaped conductor of radius `R_(1)` surrounded by an adjacent concentric layer of dielectric with permittivity `epsilon` and outside radius `R_(2)`.

Text Solution

Verified by Experts

Let us mentally inmpart a charge `q` on the conductor, then
`varhpi_(+) - varphi_(-) = int_(R_(1))^(R_(2)) (q)/(4pi epsilon_(0) epsilon r^(2)) dr + int_(R_(2))^(oo) (q)/(4pi epsilon_(0) r^(2)) dr`
`= (q)/(4pi epsilon_(0) epsilon) [(1)/(R_(1)) - (1)/(R_(2))] + (q)/(4pi epsilon_(0) (1)/(R_(2))`
`= (q)/(4pi epsilon_(0) epsilon) [((epsilon -1))/(R_(2)) + (1)/(R_(1))]`
Hence the sought capacitance,
`C = (q)/(varphi_(+) - varphi_(-)) = (q 4pi epsilon_(0) epsilon)/(q [((epsilon -1))/(R_(2)) + (1)/(R_(1))]) = (4pi epsilon_(0) epsilon R_(1))/((epsilon - 1) (R_(1))/(R_(2)) + 1)`
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Knowledge Check

  • The capacitance of an isolated conducting sphere of radius r is

    A
    `4 pi epsilon_(0)r`
    B
    `2 pi epsilon_(0)r`
    C
    ` pi epsilon_(0)r`
    D
    `8 pi epsilon_(0)r`

  • The capacitance of an isolated conducting sphere of radius R is proportional to

    A
    `R^(-1)`
    B
    `R^(2)`
    C
    `R^(-2)`
    D
    `R`

  • The capacitance of earth of radius R treating as spherical conductor is

    A
    `4piin_(0)R`
    B
    `4piR`
    C
    `in_(0)/R`
    D
    `(4piin_(0))/R`

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