Calculate the ratio of electric to gravitational force between two electrons.

To find the ratio of the electric force to the gravitational force between two electrons, we can use Coulomb's law for the electric force and Newton's law of universal gravitation for the gravitational force. Let's break this down step by step.

Understanding the Forces

The electric force between two charged particles is given by Coulomb's law:

F_electric = k * |q₁ * q₂| / r²

Where:

• F_electric is the electric force.
• k is Coulomb's constant, approximately 8.99 x 10⁹ N m²/C².
• q₁ and q₂ are the charges of the electrons, which are both approximately -1.6 x 10^-19 C.
• r is the distance between the two electrons.

The gravitational force between two masses is given by Newton's law of gravitation:

F_gravity = G * (m₁ * m₂) / r²

Where:

• F_gravity is the gravitational force.
• G is the gravitational constant, approximately 6.674 x 10^-11 N m²/kg².
• m₁ and m₂ are the masses of the electrons, which are approximately 9.11 x 10^-31 kg.
• r is the same distance between the two electrons.

Calculating the Forces

Now, let's calculate the electric and gravitational forces between two electrons at a distance r.

Electric Force Calculation

Substituting the values into Coulomb's law:

F_electric = (8.99 x 10⁹) * |(-1.6 x 10^-19) * (-1.6 x 10^-19)| / r²

This simplifies to:

F_electric = (8.99 x 10⁹) * (2.56 x 10^-38) / r²

Gravitational Force Calculation

Now, substituting the values into Newton's law:

F_gravity = (6.674 x 10^-11) * (9.11 x 10^-31) * (9.11 x 10^-31) / r²

This simplifies to:

F_gravity = (6.674 x 10^-11) * (8.27 x 10^-61) / r²

Finding the Ratio

To find the ratio of the electric force to the gravitational force, we can set up the following equation:

Ratio = F_electric / F_gravity

Substituting the expressions we derived:

Ratio = [(8.99 x 10⁹) * (2.56 x 10^-38) / r²] / [(6.674 x 10^-11) * (8.27 x 10^-61) / r²]

The r² cancels out:

Ratio = (8.99 x 10⁹ * 2.56 x 10^-38) / (6.674 x 10^-11 * 8.27 x 10^-61)

Now, calculating the numerical values:

Numerator = 8.99 x 10⁹ * 2.56 x 10^-38 ≈ 2.30 x 10^-28

Denominator = 6.674 x 10^-11 * 8.27 x 10^-61 ≈ 5.53 x 10^-71

Finally, the ratio becomes:

Ratio ≈ (2.30 x 10^-28) / (5.53 x 10^-71) ≈ 4.16 x 10⁴²

Final Thoughts

This means that the electric force between two electrons is approximately 4.16 x 10⁴² times stronger than the gravitational force. This stark difference highlights the dominance of electromagnetic interactions compared to gravitational forces at the scale of subatomic particles.