Please help. The electron beam emerging from a certain high-energy electron acce

Question

Please help.

The electron beam emerging from a certain high-energy electron accelerator has a circular cross section of radius 1.55 mm. The beam current is 7.95 mu A. Find the current density in the beam assuming it is uniform throughout. A/m2 The speed of the electrons is so close to the speed of light that their speed can be taken as 300 Mm/s with negligible error. Find the electron density in the beam. m-3 Over what time interval does Avogadro's number of electrons emerge from the accelerator? s

Explanation / Answer

a)     Assuming the beam to be in cylindrical shape with circular cross section of radius 1.55 mm

Hence the area of cross section is     a=3.14 x r2

                                                                   =3.14 x 1.55 x 1.55

                                                                  =7.54385 mm2

Current Density= Current/Area of cross section

7.95 x 10-6/7.54385 x 10-6 m2=1.05 A/m2

b)    Let the current density be X

Therefore amount of electron passing in one second is 300 x 106 x X

Amount of charge present in amount of electrons passed in one second=300 x 106 x X x 1.6 x 10-19

This charge passed in one second is equal to amount of current

Hence

300 x 106 x X x 1.6 10-19=7.95 x 10-6

X=165625 electrons/m2

c)     Total Current= 7.95 x 10-6

Hence the no of electrons passed in one second is 7.95 x 10-6/1.6 x 10-19

                                  =496875 x 108 electrons

So for Na(Avogadro no) electrons ,time required is

                                                 6.023 x 1023/4.96875 x 1013

Hence time required is 1.21 x 1010       seconds

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