A non conducting uniform solid sphere of mass m and radius R having uniform volume charge density +ρ and −ρ in two hemispherical halves as shown in figure. It is then placed on a rough non conducting horizontal plane. At t = 0 a uniform electric field E→=E0i^ N/C is switched on and the solid sphere starts rolling without sliding. The magnitude of frictional force at t = 0 is 2ρπR3E0K where K is a positive integer. Find the value of K. (Take tan53°=43 )

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A non conducting uniform solid sphere of mass m and radius R having uniform volume charge density $+ ρ$ and $- ρ$ in two hemispherical halves as shown in figure. It is then placed on a rough non conducting horizontal plane. At t = 0 a uniform electric field $\vec{E} = E_{0} \hat{i} N / C$ is switched on and the solid sphere starts rolling without sliding. The magnitude of frictional force at t = 0 is $\frac{2 ρ π R^{3} E_{0}}{K}$ where K is a positive integer. Find the value of K. (Take $\tan 53 ° = \frac{4}{3}$ )

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Detailed Solution

Charges can be assumed to be concentrated at center of mass of respective hemispheres.

Distance of COM from center of sphere is $d = \frac{3 R}{8}$ .

$q = d e n s i t y \times v o l u m e o f h e m i s p h e r e = ρ \frac{2}{3} π R^{3}$

Dipole Moment, $P = q 2 d = (ρ \frac{2}{3} π R^{3}) (\frac{3 R}{4})$

$\begin{array}{l} τ_{Q} = P E \sin 53 ° = (ρ \frac{2}{3} π R^{3}) (\frac{3 R}{4}) E_{0} \frac{4}{5} = \frac{2}{5} ρ π R^{4} E_{0} \\ B U T \\ T o r q u e, τ = I α = (\frac{7}{5} m R^{2} α) \\ \Rightarrow \frac{2}{5} ρ π R^{4} E_{0} = (\frac{7}{5} m R^{2} α) \\ \Rightarrow α = \frac{2}{7} \frac{ρ π R^{2} E_{0}}{m} \\ a n d \\ a_{c m} = R α = \frac{2}{7} \frac{ρ π R^{3} E_{0}}{m} \\ f = m a_{c m} = \frac{2}{7} ρ π R^{3} E_{0} \end{array}$