Question

The muon is a subatomic particle with the same charge as an electron but with a mass that is 207 times greater: m?=207me. Physicists think of muons as "heavy electrons." However, the muon is not a stable particle; it decays with a half-life of 1.5 ?s into an electron plus two neutrinos. Muons from cosmic rays are sometimes "captured" by the nuclei of the atoms in a solid. A captured muon orbits this nucleus, like an electron, until it decays. Because the muon is often captured into an excited orbit (n>1), its presence can be detected by observing the photons emitted in transitions such as 2?1 and 3?1.
Consider a muon captured by a carbon nucleus (Z=6). Because of its large mass, the muon orbits well inside the electron cloud and is not affected by the electrons. Thus the muon "sees" the full nuclear charge Ze and acts like the electron in a hydrogen-like ion.

Part 1: What is the orbital radius of a muon in the n=1 ground state? Note that the mass of a muon differs from the mass of an electron? Solving for r₁

Part 2: What are the speed of a muon in the n=1 ground state? Solving for v₁

Part 3: What is the wavelength of the 2?1 muon transition? solving for lambda

Part 4: How many orbits will the muon complete during 1.5 ?s? Solve for N

Answer 1

(a)4.26 x 10^-5 nm

(b)1.31 x 10^7 m/s

(c) 0.0164 nm

(d) 7.3 x 10^13 orbits

Given
mass of themuonm?=207me
mass of the electron me =9.1*10-31kg
Planck's cosntant h=6.62*10-34Js
charge of the electrone =1.6*10-31kg

(f) Yes it is a very large number even though the muon is not stable

Rydberg constant R =10.97*106/m
Permittivity constant?o=8.9*10-12C2/N-m2
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a) the radius of the muon r = (n2h2?o2)/(?m?e2)
Here n =1
Speed of the muon is
v = (e2) /( 2nh?o)
Plug in values.

4.26 x 10^-5 nm ..............ans

b) we have the formula
1/? = R [ 1/n12-1/n22]
Here n1 =1 & n2 = 2
find ? i.e wavelength of muon

1.31 x 10^7 m/s .............ans

c) Yes,
Energyof photon is
E =(hc) / ?
Plug in values.

0.0164 nm ..........ans

d) Number of orbits is given by
N = ( 1.5?s) /T
= (1.5*10-6s) /T
= (1.5*10-6s ) (m?e4/(4?o2n3h3))
Plug in values.

7.3 x 10^13 orbits..............ans