Find the ratio of the magnitude of the electric force to the gravitational force acting between two protons.

To find the ratio of the electric force to the gravitational force acting between two protons, 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

First, we need to know the formulas for both forces:

• Electric Force (F_e): According to Coulomb's law, the electric force between two charged particles is given by:

F_e = k * (|q₁ * q₂|) / r²

• Where:
• k is Coulomb's constant (approximately 8.99 x 10⁹ N m²/C²),
• q₁ and q₂ are the charges of the protons (approximately 1.6 x 10^-19 C),
• r is the distance between the two protons.
• Gravitational Force (F_g): According to Newton's law of gravitation, the gravitational force between two masses is given by:

F_g = G * (m₁ * m₂) / r²

• Where:
• G is the gravitational constant (approximately 6.67 x 10^-11 N m²/kg²),
• m₁ and m₂ are the masses of the protons (approximately 1.67 x 10^-27 kg),
• r is the distance between the two protons.

Calculating the Forces

Now, let's calculate the electric and gravitational forces acting between two protons at a distance r:

• For the electric force:

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

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

F_e = (2.30 x 10^-28) / r²

• For the gravitational force:

F_g = (6.67 x 10^-11) * (1.67 x 10^-27 * 1.67 x 10^-27) / r²

F_g = (6.67 x 10^-11) * (2.79 x 10^-54) / r²

F_g = (1.86 x 10^-64) / r²

Finding the Ratio

Now that we have both forces, we can find the ratio of the electric force to the gravitational force:

Ratio = F_e / F_g = [(2.30 x 10^-28) / r²] / [(1.86 x 10^-64) / r²]

The r² cancels out, simplifying our ratio:

Ratio = (2.30 x 10^-28) / (1.86 x 10^-64)

Calculating this gives:

Ratio ≈ 1.23 x 10³⁶

Conclusion

This means that the electric force between two protons is approximately 1.23 x 10³⁶ times stronger than the gravitational force acting between them. This stark difference illustrates why electromagnetic forces dominate at the atomic level, overshadowing gravitational forces in most practical scenarios.