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 $$2$$ρπ$$R3E0K$$ where K is a positive integer. Find the value of K. (Take$$ $$tan53$$°$$=43$$ $$)

Question

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 $$2$$ρπ$$R3E0K$$ where K is a positive integer. Find the value of K. (Take$$ $$tan53$$°$$=43$$ $$)

Infinity Learn · Accepted Answer

Charges can be assumed to be concentrated at center of mass of respective hemispheres.Distance of COM from center of sphere is $$d=3R8$$.$$q=density$$ ×volume of $$hemisphere=$$ρ$$23$$$$π$$$$R3Dipole$$ Moment, $$P=q2d=($$ρ$$23$$$$π$$$$R3)(3R4)$$τ$$Q=PEsin53$$°​$$=$$ρ$$23$$$$π$$$$R33R4E045$$​$$=25$$ρπ$$R4E0$$ BUT Torque ,τ   $$=I$$α  $$=75mR2$$α ⇒$$25$$ρπ$$R4E0=75mR2$$α ⇒α$$=27$$ρπ$$R2E0mand$$ ​$$acm=R$$α$$=27$$ρπ$$R3E0m$$​$$f=macm=27$$ρπ$$R3E0$$

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