Question

Determine the ratio of the electrostatic force to the gravitational force between a proton and an electron, FE/FG. Note: k = 8.99 × 109 N-m2/C2; G = 6.672 × 10–11 N-m2/kg2; me = 9.109 ×

Answer 1

Complete Question:

Determine the ratio of the electrostatic force to the gravitational force between a proton and electron, FE/FG.

Note : k = 8.99 × 10⁹ N.m²/C² ; G = 6.672 x 10⁻¹¹ Nm²/kg²; me = 9.109 × 10⁻³¹ kg and mp = 1.672 × 10⁻²⁷kg.

Answer:

FE/FG = 2.3 x 10³⁹

Step-by-step explanation:

According to Coulomb's law, the electrostatic force (
`F_(E)`) between two particles is given as;

`F_(E)` = k x
`(Q_1 * Q_2)/(r^2 )` --------------------(i)

Where;

k = electric constant = 8.99 x 10⁹Nm²/C²

Q₁ = the charge of particle 1

Q₂ = the charge of particle 2

r = the distance of separation between the two particles

Also, according to Newton's law of gravitational force, the gravitational force (
`F_(G)`) between two particles is given as;

`F_(G)` = G x
`(M_(1) * M_2)/(r^(2) )` --------------------(ii)

Where;

G = gravitational constant = 6.672 x 10⁻¹¹Nm²/kg²

M₁ = mass of particle 1

M₂ = mass of particle 2

r = distance of separation between the two particles

For clarity, we will calculate
`F_(E)` and
`F_(G)` separately before finding their ratio.

From the question;

The particles are a proton (particle 1) and an electron (particle 2) with the following details;

Q₁ = charge of proton = 1.6 x 10⁻¹⁹C

Q₂ = charge of electron = -1.6 x 10⁻¹⁹C

M₁ = mass of proton = mp = 1.672 × 10⁻²⁷kg

M₂ = mass of electron = me = 9.109 × 10⁻³¹kg

Substitute the values of k, Q₁ and Q₂ into equation (i) as follows;

`F_(E)` = 8.99 x 10⁹ x
`((1.6*10^(-19)) * ( -1.6* 10^(-19)))/(r^2 )`

`F_(E)` = 8.99 x 10⁹ x
`((2.56*10^(-38)))/(r^2 )` [negative sign can be discarded]

`F_(E)` =
`((23.01*10^(-29)))/(r^2 )`

Also, substitute the values of G, M₁ and M₂ into equation (ii) as follows;

`F_(G)` = 6.67 x 10⁻¹¹ x
`((1.672*10^(-27)) * (9.109* 10^(-31)))/(r^2 )`

`F_(G)` = 6.67 x 10⁻¹¹ x
`((15.23*10^(-58)))/(r^2 )`

`F_(G)` =
`((101.58*10^(-69)))/(r^2 )`

The ratio
`F_(E)` /
`F_(G)` is therefore;

`F_(E)` /
`F_(G)` =
`((23.01*10^(-29)))/(r^2 )` /
`((101.58*10^(-69)))/(r^2 )`

`F_(E)` /
`F_(G)` =
`((23.01*10^(-29)))/((101.58*10^(-69)))`

`F_(E)` /
`F_(G)` =
`{(0.23*10^(40))}`

`F_(E)` /
`F_(G)` =
`{(2.3*10^(39))}`

Therefore, the ratio is 2.3 x 10³⁹

Answer 2

`(F_e)/(F_g) = 2.3 * 10^(18)`

Step-by-step explanation:

The gravitational force is given by Newton's Law of Gravity:

`F_g = (Gm_1m_2)/(r^2)`

The electrostatic force is given by Coulomb's Law:

`F_e = (kq_1q_2)/(r^2)`

The ratio between these two forces is

`(F_e)/(F_g) = ((kq_1q_2)/(r^2))/((Gm_1m_2)/(r^2)) = (kq_1q_2)/(Gm_1m_2) = ((8.99* 10^(-12))(1.6* 10^(-19))(1.6 * 10^(-19)))/((6.67* 10^(-11))(9.1 * 10^(-31))(1.6 * 10^(-27))) = 2.3 * 10^(18)`