A spherical capacitor has an inner sphere of radius 12 cm and an outer sphere of radius 13 cm. The outer sphere is earthed and the inner sphere is given a charge of 2.5 muC. The space between the concentric spheres is filled with a liquid of dielectric constant 32. <br> (a) Determine the capacitance of the capacitor. (b) What is the potential of the inner sphere ? <br> (c) Compare the capacitance of this capacitor with that of an isolated sphere of radius 12 cm.Explain why the later is much smaller ?

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Answer Text

Solution :
Radius of the inner sphere, `r_(2)=12 cm =0.12 m ` <br> Radius of the outer sphere , `r_(1)=13 cm =0.13 m ` <br> charge on the inner sphere , `q=2.5 muC=2.5xx10^(-6)C` <br> Dielectric constant of a liquid , `in_(r)=32` <br> (a) capcitances of the capacitors is given by the relation <br> `C=(4pi in_(0)in_(r)r_(1)r_(2))/(r_(1)-r_(2))` <br> where, `in_(0)=`permitivity of free space =`8.85xx10^(-12)C^(2)N^(-4)m^(-2)` <br> `1/(4piin_(0))=9xx10^(9)Nm^(2)C^(-2)` <br> `:. C=(32xx0.12xx0.13)/(9xx10^(9)xx(0.13-0.12))` <br> `=5.5xx10^(-9)F` <br> Hence, the capacitance of the capacitors is approximately `5.5xx10^(-9) F`. (b)potential of the inner sphere is given by <br> `V=q/C` <br> `=(2.5xx10^(-6))/(5.5xx10^(-6))=4.5xx10^(2)V` <br> Hence, the potential of the inner sphere is `4.5xx10^(2)V` <br> (c) radius of an isolated sphere `, r=12xx10^(-2)m` <br> capcaitance of the sphere is given by the relation. <br> `C'=4pi in_(0)r` <br> `=4pixx8.85xx10^(-12)xx12xx10^(-12)` <br> `1.33xx10^(-11)F` <br> The capacitance of the isolated sphere is less in comparison to the concentric spheres. thie is because the outer sphere of the concentric sphers is eartherd . hence, the potential difference is less and the capacitances is more than the issolated sphere.

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