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NTA Abhyas 2020
Exam
NTA Abhyas
Year
2020
Subject
Physics
Chapter
Mechanical Properties of Solids

NTA Abhyas 2020 · Physics · Mechanical Properties of Solids

A solid sphere made out of metal having a bulk modulus of 2 × 10 ^...

A solid sphere made out of metal having a bulk modulus of 2×1011Nm−22 \times {10}^{11}{Nm}^{- 2} is charged uniformly to a surface charge density σ=αε0Cm−2\sigma = \sqrt{\alpha{ε}_{0}}{Cm}^{- 2}, where ε0{ε}_{0} is the permittivity of free space and α\alpha (magnitude = 11) is a positive constant of appropriate dimensions. If the volume strain produced in the sphere after charging is x×10−11x \times {10}^{- 11}, then the value of xx is

Correct answer

0.25

Step-by-step solution

Bulk modulus is given by

B=−\DeltaP\DeltaVVB = \dfrac{- \DeltaP}{\dfrac{\DeltaV}{V}}

The outward electrostatic pressure is

Pelectrostatic=σ22ε0=−\DeltaP{P}_{electrostatic} = \dfrac{{\sigma}^{2}}{2{ε}_{0}} = - \DeltaP

⇒\DeltaVV=σ22ε0B⇒ \dfrac{\DeltaV}{V} = \dfrac{\dfrac{{\sigma}^{2}}{2{ε}_{0}}}{B}

⇒\DeltaVV=(αε0)22ε02×1011=0.25×10−11⇒ \dfrac{\DeltaV}{V} = \dfrac{\dfrac{{\left(\sqrt{\alpha{ε}_{0}}\right)}^{2}}{2{ε}_{0}}}{2 \times {10}^{11}} = 0 . 25 \times {10}^{- 11}

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