Question

Suppose that an electron and a proton are placed at a distance of 1.2nm away from each other (about 10 times the radius of a hydrogen atom).

(a) How much is the Coulomb force between the electron and the proton?

(b) Under this Coulomb force alone, how much is the acceleration of the electron towards the proton (in m/s2 )? (Google for the missing information.)

(c) If the electron is instead placed on the edge of the supermassive black hole Sagittarius A* (about 22 million km away from the center), how much is the gravitational acceleration of electron towards the black hole (in m/s2 )? (Google for the missing information about Sagittarius A*.)

(d) For this electron, how does the gravitational influence of the black hole compare to the electric influence of the proton?

Answer 1

mass of a proton,m_p =1.67 × 10^-27 kg

mass of an electron ,m_e =9.11 × 10^-31 kg

charge on a proton(q_p) = +1.602 × 10^-19 C

charge on an electron(q_e) =-1.602 × 10^-19 C

mass of Sagittarius A* ,m_s= 4.1 million solar masses = 4.1x10⁶x1.989 × 10³⁰ kg = 8.155x10³⁶kg

G=  6.67 x 10^-11 N m²/kg²

k =9.0 x10⁹ N

F_E =Coulomb force

F_G =Gravitational force

a= acceleration

a.

|F|_E =k|q_p||q_e|/r²

= (9.0 x10⁹)(1.602 × 10^-19)(1.602 × 10^-19)/(1.2x10^-9)²

= 16.04x10^-11 N

b.

a =F/m_e = 16.04x10^-11 N / 9.11 × 10^-31kg = 1.76x10²⁰m/s²

c.

Gravitational force between the electron and sagittarius A* = Gm_em_s/r²

F_G= (6.67 x 10^-11)(9.11 × 10^-31)(8.155x10³⁶)/(22x10⁹)²

=1.024 x 10^-24N

a= Gravitational force/mass of electron = 1.024 x 10^-24N / 9.11 × 10^-31kg = 0.1124 x 10⁷ m/s²

d.

accn due to F_E / due to F_G = 1.76x10²⁰/0.1124 x 10⁷ =15.66 x 10¹³

So, acceleration due to the electric force between electron and proton is about 15.66 x 10¹³ times the acceleration due to gravitational force between the electron and sagittarius A*.