A sphere of radius 10 mm that carries a charge of 8nC is attached at the end of an insulating string and is whirled in a circle. The rotation frequency is 100? rad/s. What average current does this rotating charge represent ?

Surface area of the sphere is 4πr2=4×π×10×10-32=0.125 m2. For a conducting sphere all charges will be on surface area, so surface charge density is δ=8×10-90.125=6.4×10-8c/m2. Current for any spinning sphere with constant charge density is given by I=ωrsinθδ for a particular point on the surface. Now to get the total current we have to integrate the above equation from θ=0 to θ=π. Now integrating we have
Itot=0πωrsinθδ=ωrδ0πsinθ=2ωrδ
Now we have r=10mm=.01m and ω=100rad/s, so we have
I=2×0.01×100×6.4×10-8=1.28×10-7A

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