Ask questions, share knowledge, and connect with a vibrant community on IDNLearn.com. Our platform provides detailed and accurate responses from experts, helping you navigate any topic with confidence.
Sagot :
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]
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]
We greatly appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. IDNLearn.com is your source for precise answers. Thank you for visiting, and we look forward to helping you again soon.
