To calculate the mass of an electron using the gravitational force between two electrons that are \(0.98\) meters apart, you can use Newton's law of universal gravitation. The formula for the gravitational force \(F\) between two masses \(m_1\) and \(m_2\) separated by a distance \(r\) is given by:
\[ F = G \frac{m_1 m_2}{r^2} \]
Where:
- \(F\) is the gravitational force between the masses,
- \(G\) is the gravitational constant \(6.674 \times 10^{-11} , \text{m}^3 , \text{kg}^{-1} , \text{s}^{-2}\),
- \(m_1\) and \(m_2\) are the masses of the two electrons,
- \(r\) is the distance between the centers of the two masses (in this case, \(0.98\) meters).
Since both masses are equal (mass of an electron), we can denote the mass of an electron as \(m\):
\[ F = G \frac{m^2}{r^2} \]
Steps to calculate the mass of the electron:
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Calculate the gravitational force between the two electrons. The gravitational force \(F\) between two electrons is extremely small and is calculated using the known mass of an electron, approximately \(9.109 \times 10^{-31}\) kg.
\[ r = 0.98 , \text{m} \]
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Substitute the known values into the equation. First, we need to find \(F\):
\[ F = G \frac{(9.109 \times 10^{-31} , \text{kg})^2}{(0.98 , \text{m})^2} \]
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Calculate \(F\):
\[ F = (6.674 \times 10^{-11} \text{ m}^3 \text{ kg}^{-1} \text{ s}^{-2}) \cdot \frac{(9.109 \times 10^{-31})^2}{(0.98)^2} \]
First, let's calculate \((9.109 \times 10^{-31})^2\):
\[ (9.109 \times 10^{-31})^2 \approx 8.27 \times 10^{-61} , \text{kg}^2 \]
Next, calculate \(0.98^2 \approx 0.9604 , \text{m}^2\).
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Now substitute back into the force equation:
\[ F \approx 6.674 \times 10^{-11} \cdot \frac{8.27 \times 10^{-61}}{0.9604} \]
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Finally calculate \(F\):
\[ F \approx 6.674 \times 10^{-11} \cdot 8.61 \times 10^{-61} \approx 5.75 \times 10^{-71} , \text{N} \]
This value \(F\) represents the gravitational force between two electrons separated by \(0.98\) meters. The gravitational force between electrons at this distance is very small and difficult to measure due to various types of forces (such as electromagnetic) being significantly stronger.
Conclusion
While we set out to calculate the gravitational force between two electrons and used that to infer the mass of an electron, the exercise largely serves to illustrate how small the gravitational force between subatomic particles is compared to other fundamental forces. The known mass of an electron remains approximately \(9.109 \times 10^{-31}\) kg, and this calculation highlights the challenges of measuring gravitational interactions at such small scales.
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