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QJEEM20Pha41c: JEE Main 2020 Physics - Class 12 JEEM20Pha41c Solution: A charge Q is distributed over two concentric conducting thin…

A charge Q is distributed over two concentric conducting thin spherical shells radii r and R (R > r). If the surface charge densities on the two shells are equal, the electric potential at the common centre is : JEE Main 2020 (Online) 2nd September Evening Slot Physics - Electrostatics Question 152 English

Multiple Choice Options

  • A 14πε0(R+r)(R2+r2){1 \over {4\pi {\varepsilon _0}}}{{\left( {R + r} \right)} \over {\left( {{R^2} + {r^2}} \right)}} Q
  • B 14πε0(R+r)2(R2+r2)Q{1 \over {4\pi {\varepsilon _0}}}{{\left( {R + r} \right)} \over {2\left( {{R^2} + {r^2}} \right)}}Q
  • C 14πε0(R+2r)Q2(R2+r2){1 \over {4\pi {\varepsilon _0}}}{{\left( {R + 2r} \right)Q} \over {2\left( {{R^2} + {r^2}} \right)}}
  • D 14πε0(2R+r)(R2+r2)Q{1 \over {4\pi {\varepsilon _0}}}{{\left( {2R + r} \right)} \over {\left( {{R^2} + {r^2}} \right)}}Q

Step-by-Step Verified Solution

Let the charges on inner and outer spheres are Q1 and Q2.

∴\therefore Q1 + Q2 = Q ...(1)

Since charge density ' σ\sigma ' is same for both spheres, so

σ\sigma = Q14πr2=Q24πR2{{{Q_1}} \over {4\pi {r^2}}} = {{{Q_2}} \over {4\pi {R^2}}}

⇒\Rightarrow Q1Q2=r2R2{{{Q_1}} \over {{Q_2}}} = {{{r^2}} \over {{R^2}}} ....(2)

From (1) and (2),

Q2r2R2+Q2=Q{{{Q_2}{r^2}} \over {{R^2}}} + {Q_2} = Q

⇒\Rightarrow Q2 = QR2(r2+R2){{Q{R^2}} \over {\left( {{r^2} + {R^2}} \right)}}

∴\therefore Q1 = r2R2.QR2(r2+R2){{{r^2}} \over {{R^2}}}.{{Q{R^2}} \over {\left( {{r^2} + {R^2}} \right)}} = Qr2(r2+R2){{Q{r^2}} \over {\left( {{r^2} + {R^2}} \right)}}

Potential at centre 'O' = kQ1r+kQ2R{{k{Q_1}} \over r} + {{k{Q_2}} \over R}

= k[Qr2r(r2+R2)+QR2R(r2+R2)]k\left[ {{{Q{r^2}} \over {r\left( {{r^2} + {R^2}} \right)}} + {{Q{R^2}} \over {R\left( {{r^2} + {R^2}} \right)}}} \right]

= kQ(r+R)(r2+R2){{{kQ\left( {r + R} \right)} \over {\left( {{r^2} + {R^2}} \right)}}}

= 14πε0(R+r)(R2+r2){1 \over {4\pi {\varepsilon _0}}}{{\left( {R + r} \right)} \over {\left( {{R^2} + {r^2}} \right)}} Q

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