2.31 (a) Two large conducting spheres carrying charges ${\mathit{Q}}_1$ and ${\mathit{Q}}_2$ are brought close to each other. Is the magnitude of electrostatic force between them exactly given by ${\mathit{Q}}_1{\mathit{Q}}_2$ /4 $\mathit{\pi }{\mathit{\epsilon }}_0{\mathit{r}}^2$ , where r is the distance between their centres?

(b) If Coulomb’s law involved 1/ ${\mathit{r}}^3$ dependence (instead of 1/ ${\mathit{r}}^2$ ), 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).

<p><strong>2.31 </strong>The force between two conducting spheres is not exactly given by the expression&nbsp;<span title="Click to copy mathml"><math><msub><mrow><mrow><mi mathvariant="bold-italic">Q</mi></mrow></mrow><mrow><mrow><mn>1</mn></mrow></mrow></msub><msub><mrow><mrow><mi mathvariant="bold-italic">Q</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msub></math></span>&nbsp;/4&nbsp;<span title="Click to copy mathml"><math><mi mathvariant="bold-italic">? </mi><msub><mrow><mrow><mi mathvariant="bold-italic">? </mi></mrow></mrow><mrow><mrow><mn>0</mn></mrow></mrow></msub><msup><mrow><mrow><mi mathvariant="bold-italic">r</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msup></math></span>&nbsp;, because there is non-uniform charge distribution on the spheres.</p><p>Gauss's law will not be true, if Coulomb's law involved&nbsp;<span title="Click to copy mathml"><math><mfrac><mrow><mrow><mn>1</mn></mrow></mrow><mrow><mrow><msup><mrow><mrow><mi>r</mi></mrow></mrow><mrow><mrow><mn>3</mn></mrow></mrow></msup></mrow></mrow></mfrac></math></span>&nbsp;dependence, instead of&nbsp;<span title="Click to copy mathml"><math><mfrac><mrow><mrow><mn>1</mn></mrow></mrow><mrow><mrow><msup><mrow><mrow><mi>r</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msup></mrow></mrow></mfrac></math></span>&nbsp;, on r</p><p>Yes. If a small test charge is released at rest at a point in an electrostatic field configuration, then it will travel along the field lines passing through the point, only if the field lines are straight. This is because the field lines give the direction of acceleration and not the velocity.</p><p>Whenever the electron completes an orbit, either circular or elliptical, the work done by the field of a nucleus is zero.</p><p>No. Electric field is discontinuous across the surface of a charged conductor. However, electric potential is continuous.</p><p>The capacitance of a single conductor is considered as a parallel plate capacitor with one of its two plates at infinity.</p><p>Water has an unsymmetrical space as compared to mica. Since it has a permanent dipole moment, it has a greater dielectric constant than mica.</p>

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