A source in the form of a metal sphere of diameter $$10-3$$ m emits electrons at a constant rate of $$6.25$$ x $$1010$$ particles per second. If the source is electrically insulated, how long (in$$ μ$$s) will it take for its potential to rise by $$1.0$$ volt, assuming that $$80$$% of emitted electrons escape from the surface.

Question

A source in the form of a metal sphere of diameter $$10-3$$ m emits electrons at a constant rate of $$6.25$$ x $$1010$$ particles per second. If the source is electrically insulated, how long (in$$ μ$$s) will it take for its potential to rise by $$1.0$$ volt, assuming that $$80$$% of emitted electrons escape from the surface.

Infinity Learn · Accepted Answer

Capacitance of $$sphereC=4$$πε$$0R=0.5$$×$$10$$−$$39$$×$$109=118$$×$$10$$−$$12$$ FRate of escape of charge from $$surface=80100$$×$$6.25$$×$$1010$$×$$1.6$$×$$10$$−$$19=8$$×$$10$$−$$9$$ $$C/s20$$% remained with source only. so charge developed is same as electrons escaped.therefore,  $$q=8$$×$$10-9t$$ and $$q=CV$$⇒$$8$$×$$10$$−$$9t=118$$×$$10$$−$$12$$×$$1$$⇒$$t=10$$−$$128$$×$$10$$−$$9$$×$$18=10$$−$$3144=6.94$$ μs

Electrostatic potential

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