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Electrostatics question

2019 · 10 Jan · Shift 1 · Q46
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Electrostatics question

2019 · 10 Jan · Shift 1 · Q46

JEE MainPhysicsElectrostaticsMCQ+4 / −1
A charge Q is distributed over three concentric spherical shells of radii a, b, c (a < b < c) such that their surface charge densities are equal to one another. The total potential at a point at distance r from their common centre, where r < a, would be -
  1. A
    Q(a2+b2+c2)4πε0(a3+b3+c3){{Q\left( {{a^2} + {b^2} + {c^2}} \right)} \over {4\pi {\varepsilon _0}\left( {{a^3} + {b^3} + {c^3}} \right)}}4πε0​(a3+b3+c3)Q(a2+b2+c2)​
  2. B
    Q4πε0(a+b+c){Q \over {4\pi {\varepsilon _0}\left( {a + b + c} \right)}}4πε0​(a+b+c)Q​
  3. C
    Q12πε0ab+bc+caabc{Q \over {12\pi {\varepsilon _0}}}{{ab + bc + ca} \over {abc}}12πε0​Q​abcab+bc+ca​
  4. D
    Q(a+b+c)4πε0(a2+b2+c2){{Q\left( {a + b + c} \right)} \over {4\pi {\varepsilon _0}\left( {{a^2} + {b^2} + {c^2}} \right)}}4πε0​(a2+b2+c2)Q(a+b+c)​
View written solutionFree

Correct answer: D

  1. Let the common surface charge density be σ\sigmaσ.

    Since charge is distributed on three concentric spherical shells of radii a,b,ca, b, ca,b,c with equal surface charge density, the charges on the shells are:

    qa=4πa2σ,qb=4πb2σ,qc=4πc2σq_a = 4\pi a^2 \sigma, \quad q_b = 4\pi b^2 \sigma, \quad q_c = 4\pi c^2 \sigmaqa​=4πa2σ,qb​=4πb2σ,qc​=4πc2σ

  2. Use total charge QQQ.

    Q=qa+qb+qc=4πσ(a2+b2+c2)Q = q_a + q_b + q_c = 4\pi \sigma (a^2 + b^2 + c^2)Q=qa​+qb​+qc​=4πσ(a2+b2+c2)

    Hence,

    σ=Q4π(a2+b2+c2)\sigma = \frac{Q}{4\pi (a^2+b^2+c^2)}σ=4π(a2+b2+c2)Q​

  3. Potential at a point inside a spherical shell.

    For a spherical shell of radius RRR carrying charge qqq, the potential at any interior point is constant:

    V=14πε0qRV = \frac{1}{4\pi \varepsilon_0}\frac{q}{R}V=4πε0​1​Rq​

    Since r<ar<ar<a, the point lies inside all three shells.

    Therefore total potential is:

    V=14πε0(qaa+qbb+qcc)V = \frac{1}{4\pi \varepsilon_0}\left(\frac{q_a}{a} + \frac{q_b}{b} + \frac{q_c}{c}\right)V=4πε0​1​(aqa​​+bqb​​+cqc​​)

  4. Substitute qa,qb,qcq_a, q_b, q_cqa​,qb​,qc​.

    V=14πε0(4πa2σa+4πb2σb+4πc2σc)V = \frac{1}{4\pi \varepsilon_0}\left(\frac{4\pi a^2\sigma}{a} + \frac{4\pi b^2\sigma}{b} + \frac{4\pi c^2\sigma}{c}\right)V=4πε0​1​(a4πa2σ​+b4πb2σ​+c4πc2σ​)

    V=14πε0⋅4πσ(a+b+c)V = \frac{1}{4\pi \varepsilon_0} \cdot 4\pi \sigma (a+b+c)V=4πε0​1​⋅4πσ(a+b+c)

    V=σ(a+b+c)ε0V = \frac{\sigma (a+b+c)}{\varepsilon_0}V=ε0​σ(a+b+c)​

  5. Now substitute σ\sigmaσ.

    V=1ε0⋅Q4π(a2+b2+c2)(a+b+c)V = \frac{1}{\varepsilon_0} \cdot \frac{Q}{4\pi (a^2+b^2+c^2)}(a+b+c)V=ε0​1​⋅4π(a2+b2+c2)Q​(a+b+c)

    V=Q(a+b+c)4πε0(a2+b2+c2)V = \frac{Q(a+b+c)}{4\pi \varepsilon_0 (a^2+b^2+c^2)}V=4πε0​(a2+b2+c2)Q(a+b+c)​

  6. Compare with options.

    This matches Option D:

    Q(a+b+c)4πε0(a2+b2+c2)\boxed{\frac{Q(a+b+c)}{4\pi \varepsilon_0 (a^2+b^2+c^2)}}4πε0​(a2+b2+c2)Q(a+b+c)​​

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