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Question
Consider the following very rough model of a beryllium atom. The nucleus has four protons and four neutrons confined to a small volume of radius 10−15 m. The two 1 selectrons make a spherical charge cloud at an average distance of 1⋅3 ×10−11 m from the nucleus, whereas the two 2 s electrons make another spherical cloud at an average distance of 5⋅2 × 10−11 m from the nucleus. Find three electric fields at (a) a point just inside the 1 s cloud and (b) a point just inside the 2 s cloud.
Solution
(a)
Let us consider the three surfaces as three concentric spheres A, B and C.
Let us take q = 1.6× 10-19 C .
Sphere A is the nucleus; so, the charge on sphere A, q1 = 4q
Sphere B is the sphere enclosing the nucleus and the 2 1s electrons; so charge on this sphere, q2 = 4q -2q = 2q
Sphere C is the sphere enclosing the nucleus and the 4 electrons of Be; so, the charge enclosed by this sphere,
q3 = 4q -4q = 0
Radius of sphere A, r1 = 10-15 m
Radius of sphere B, r2 = 1.3 × 10-11 m
Radius of sphere C, r3 = 5.2 × 10-11 m
As the point 'P' is just inside the spherical cloud 1s, its distance from the centre
x = 1.3 × 10-11 m
Electric field,
`"E" = q/(4 pi ∈_0 "x"^2)`
Here, the charge enclosed is due to the charge of the 4 protons inside the nucleus . So,
`"E" = (4 xx (1.6 xx 10^-19))/(4 xx 3.14 xx (8.85 xx 10^-12)xx ( 1.3 xx 10^-11)^2 ) `
E = 3.4× 1013 N/C
(b)
For a point just inside the 2s cloud, the total charge enclosed will be due to the 4 protons and 2 electrons. Charge enclosed,
qen = 2q = 2× (1.6 ×10-19) C
Hence, electric field,
`"E" = ("q"_"en")/ (4 pi ∈_0"x"^2)`
x = 5.2× 10-11 m
`"E" = (2 xx (1.6 xx 10^-19))/(4 xx 3.14 xx ( 8.85 xx 10^-12) xx ( 5.2 xx 10^-11)^2)`
E = 1.065 × 1012 N/C
Thus , E =1.1× 1012 N/C
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