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If Coulomb's law involved 1/r^3 instead ...

If Coulomb's law involved `1/r^3` instead of `1/r^2`, would Gauss's law still be true?

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No, Gauss.s law will not be true in the case.
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Answer carefully: (a) Two large conducting spheres carrying charges Q_(1) and Q_(2) are brought close to each other. Is the magnitude of electrostatic force between them exactly given by Q_(1) Q_(2)//4pi epsilon_(0) r_(2) , where r is the distance between their centers? (b) If Coulomb’s law involved 1//r^(3) dependence (instead of 1//r^(2) ), would Gauss’s law be still true ? (c) A small test charge is released at rest at a point in an electrostatic field configuration. Will it travel along the field line passing through that point? (d) What is the work done by the field of a nucleus in a complete circular orbit of the electron? What if the orbit is elliptical? (e) We know that electric field is discontinuous across the surface of a charged conductor. Is electric potential also discontinuous there? (f) What meaning would you give to the capacitance of a single conductor? (g) Guess a possible reason why water has a much greater dielectric constant (= 80) than say, mica (= 6).

Gauss's law part-1

Gauss's law part-2

If It is 1/r^2.5 in place of 1/r^2 , would Gauss's law still hold?

Which of the following statement(s) is/are correct? If the electric field due to a point charge varies as r^(-2.5) instead of r^(-2),then the Gauss law will still be valid. The Gauss law can be used to calculate the field distribution around an electric dipole. If he electric field between two point charges is zero somewhere,then the sign of the two charges is the same. The work done by the external force is moving a unit positive charge from point A at potential "V_(A)" to point B at potential "V_(B)" is,(V_(B)-V_(A))

Assertion: Column's law can be derived from Gauss's law. Reason: Gauss's law can be derived from Coulombs law.

The Bohr model for the H-atom relies on the Coulomb's law of electrostatics . Coulomb's law has not directly been varified for very short distances of the order of angstroms. Suppos-ing Coulomb's law between two oppsite charge +q_(1),-q_(2) is modified to |vec(F)|=(q_(1)q_(2))/((4piepsilon_(0))r^(2))1/r^(2),rgeR_(0) =(q_(1)q_(2))/((4piepsilon_(0))r^(2))1/R_(0)^(2)(R_(0)/r)^(epsilon), rleR_(0) Calculate in such a case , the ground state enenergy of H-atom , if epsilon 0.1,R_(0)=1Å