Question

In the most common isotope of Hydrogen the nucleus is made out of a single proton. When this Hydrogen atom is neutral, a single electron orbits around the nucleus. What is the ratio of the electric force and the gravitational force acting between the proton and the electron, when they are 4.53 angstrom away from each other? (Possibly useful constants: Coulomb constant: k = 8.9876×109 N*m2/C2, Universal Gravitational constant: G = 6.6726×10-11 N*m2/kg2, elementary charge: e = 1.6022×10-19 C, electron mass: me = 9.1094×10-31 kg, proton mass: mp = 1.6726×10-27 kg. You might also find the specific charges i.e. charge per mass ratios of the electron and the proton useful: electron: e/me = 1.7588×1011 C/kg; proton: e/mp = 9.5788×107 C/kg.)

Answer 1

The ratio of the electric force to the gravitational force acts as a comparison between the strength of Coulomb's law and Newton's law of universal gravitation at a given distance between an electron and a proton in a neutral hydrogen atom. The distance factor cancels out in the ratio, emphasizing the dominance of the electrostatic force over the gravitational force.

Step-by-step explanation:

The question asks to compare the magnitude of the electrostatic force, due to Coulomb's law, to the gravitational force, due to Newton's law of universal gravitation, between a single proton and electron at a distance of 4.53 angstroms (4.53 × 10^-10 meters) when they are in a neutral hydrogen atom.

To find the electric force (electric force), we use Coulomb's law: Fe = k × (e × e) / r^2, where k is the Coulomb constant, e is the elementary charge, and r is the separation distance. For the gravitational force (gravitational force), we use Newton's law: Fg = G × (me × mp) / r^2, where G is the universal gravitational constant, me is the electron mass, mp is the proton mass, and r is the same separation distance. By taking the ratio of these two forces, the distance squared cancels out, highlighting the significant discrepancy between the forces, with the Coulomb force being profoundly greater.

Answer 2

The ratio of electric force to the gravitational force is
`2.27* 10^(39)`

Step-by-step explanation:

It is given that,

Distance between electron and proton,
`r=4.53\ A=4.53* 10^(-10)\ m`

Electric force is given by :

`F_e=k(q_1q_2)/(r^2)`

Gravitational force is given by :

`F_g=G(m_1m_2)/(r^2)`

Where

`m_1` is mass of electron,
`m_1=9.1* 10^(-31)\ kg`

`m_2` is mass of proton,
`m_2=1.67* 10^(-27)\ kg`

`q_1` is charge on electron,
`q_1=-1.6* 10^(-19)\ kg`

`q_2` is charge on proton,
`q_2=1.6* 10^(-19)\ kg`

`(F_e)/(F_g)=(kq_1q_2)/(Gm_1m_2)`

`(F_e)/(F_g)=(9* 10^9* (1.6* 10^(-19))^2)/(6.67* 10^(-11)* 9.1* 10^(-31)* 1.67* 10^(-27))`

`(F_e)/(F_g)=2.27* 10^(39)`

So, the ratio of electric force to the gravitational force is
`2.27* 10^(39)`. Hence, this is the required solution.