Question

If an electron is confined within a nucleus whose diameter is $10^{-14}m$.Estimate its minimum its kinetic energy and coulomb energy.

Answer 1

Let number of protons in nucleus is Z.

according to Bohr's law,

electrostatic force is balanced by centripetal force

e.g., k(Ze)(e)/r² = mv²/r

or, KZe²/r = mv²

or, KZe²/2r = 1/2 mv² = kinetic energy

here it is clear that kinetic energy will be minimum when Z = 1

so, minimum kinetic energy = Ke²/2r

here, r = 10^-14/2 = 0.5 × 10^-14 m , e = 1.6 × 10^-19 C and K = 9 × 10^9 Nm²/C²

kinetic energy = 9 × 10^9 × (1.6 × 10^-19)²/2(0.5 × 10^-14)

= 9 × 10^9 × 2.56 × 10^-38/(10^-14)

= 9 × 2.56 × 10^-15

= 23.04 × 10^-15 J

= 2.304 × 10^-14 J

we know potential energy = -2 × kinetic energy
so, potential energy = - 2 × 2.304 × 10^-14

= 4.608 × 10^-14 J