To compare the magnitudes of the electromagnetic and gravitational forces between two electrons separated by a distance of 2.00 meters, we need to calculate both forces and then find the ratio of the electromagnetic force to the gravitational force. Here are the steps:
1. Constants and Given Data:
- Mass of an electron, [tex]\( m = 9.11 \times 10^{-31} \, \text{kg} \)[/tex]
- Charge of an electron, [tex]\( e = 1.61 \times 10^{-19} \, \text{C} \)[/tex]
- Distance between electrons, [tex]\( d = 2.00 \, \text{m} \)[/tex]
- Coulomb's constant, [tex]\( k_e = 8.99 \times 10^9 \, \text{N m}^2 \text{C}^{-2} \)[/tex]
- Gravitational constant, [tex]\( G = 6.67430 \times 10^{-11} \, \text{N m}^2 \text{kg}^{-2} \)[/tex]
2. Calculating the Electromagnetic Force [tex]\( F_e \)[/tex]:
[tex]\[
F_e = \frac{k_e \cdot e^2}{d^2}
\][/tex]
- Substituting the given values:
[tex]\[
F_e = \frac{8.99 \times 10^9 \, \text{N m}^2 \text{C}^{-2} \cdot (1.61 \times 10^{-19} \, \text{C})^2}{(2.00 \, \text{m})^2}
\][/tex]
- The calculated value for [tex]\( F_e \)[/tex] rounded to two decimal places is:
[tex]\[
F_e \approx 5.83 \times 10^{-29} \, \text{N}
\][/tex]
3. Calculating the Gravitational Force [tex]\( F_g \)[/tex]:
[tex]\[
F_g = \frac{G \cdot m^2}{d^2}
\][/tex]
- Substituting the given values:
[tex]\[
F_g = \frac{6.67430 \times 10^{-11} \, \text{N m}^2 \text{kg}^{-2} \cdot (9.11 \times 10^{-31} \, \text{kg})^2}{(2.00 \, \text{m})^2}
\][/tex]
- The calculated value for [tex]\( F_g \)[/tex] rounded to two decimal places is:
[tex]\[
F_g \approx 1.38 \times 10^{-71} \, \text{N}
\][/tex]
4. Calculating the Ratio [tex]\( \frac{F_e}{F_g} \)[/tex]:
[tex]\[
\frac{F_e}{F_g} = \frac{5.83 \times 10^{-29} \, \text{N}}{1.38 \times 10^{-71} \, \text{N}}
\][/tex]
- This results in:
[tex]\[
\frac{F_e}{F_g} \approx 4.21 \times 10^{42}
\][/tex]
Summarizing all the rounded calculations, we have:
[tex]\[
\begin{aligned}
F_e & = 5.83 \times 10^{-29} \, \text{N} \\
F_g & = 1.38 \times 10^{-71} \, \text{N} \\
\frac{F_e}{F_g} & = 4.21 \times 10^{42}
\end{aligned}
\][/tex]
So, the final results are:
[tex]\[
\begin{aligned}
F_e & = 5.83 \times 10^{-29} \, \text{N} \\
F_g & = 1.38 \times 10^{-71} \, \text{N} \\
\frac{F_e}{F_g} & = 4.21 \times 10^{42}
\end{aligned}
\][/tex]
