A radioactive source in the form of metal sphere of diameter 10−3 m emits beta particle at a constant rate of 6.25×1010 particles per second. If the source is electrically insulated, how long will it take for its potential to rise by 1.0 volt, assuming that 80% of the emitted beta particles escape from the source? 

[a] Let t = time for the potential of metal sphere to rise by one volt. Now β− particles emitted in this time =(6.25×1011)×t Number of β-particles escaped in this time =(80/100)×(6.25×1010)t=5×1010t ∴  Charge acquired by the sphere in t sec. Q=(5×1010t)×(1.6×10−19)=8×10−19t    ?.(i) (∵ emission of β -particle lends to a charge eon metal sphere) The capacitance C of a metal sphere is given by C=4πε0×r =(19×109)×(10−32)=10−1218       farad   ?.(ii) we know that Q=C×V {Here V=1 volt} ∴(8×10−9)t=(10−1218)×1 Solving it for t, we get t=6.95 μsec.