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Chapter 16: Problem 74

Consider two protons (charge \(+e\) ), separated by a distance of $2.0 \times 10^{-15} \mathrm{m}$ (as in a typical atomic nucleus). The electric force between these protons is equal in magnitude to the gravitational force on an object of what mass near Earth's surface?

Short Answer

Expert verified
Answer: The mass of the object is approximately \(9.244 \times 10^{-9} \; kg\).

Step by step solution

01

Find the electric force between the two protons

To find the electric force between the protons, we can use the Coulomb's Law formula: \(F_{electric} = k \frac{q_1 q_2}{r^2}\) where: - \(F_{electric}\) is the electric force. - \(k\) is the Coulomb's constant (\(8.9875 \times 10^{9} Nm^2/C^2\)). - \(q_1\) and \(q_2\) are the charges of the protons (+\(e\) each, with \(e = 1.6 \times 10^{-19} C\)). - \(r\) is the distance between the protons (\(2.0 \times 10^{-15} m\)). Now we will substitute the values into the equation and calculate the electric force: \(F_{electric} = k \frac{(+e)(+e)}{(2.0 \times 10^{-15})^2}\)
02

Calculate the electric force

Substitute the values of \(k\) and \(e\) into the equation: \(F_{electric} = (8.9875 \times 10^{9}) \frac{(1.6 \times 10^{-19})(1.6 \times 10^{-19})}{(2.0 \times 10^{-15})^2}\) Compute the electric force value: \(F_{electric} \approx 9.067 \times 10^{-8} N\)
03

Use the gravitational force formula to find the mass of the object

Now that we have the electric force, we can equate it to the gravitational force to find the mass of the object: \(F_{electric} = F_{gravitational}\) The gravitational force formula is: \(F_{gravitational} = m g\) where: - \(F_{gravitational}\) is the gravitational force. - \(m\) is the mass of the object. - \(g\) is the acceleration due to gravity (\(9.81 \; m/s^2\)). Now we will equate the electric and gravitational forces and solve for the mass: \(9.067 \times 10^{-8} N = m(9.81 \; m/s^2)\)
04

Calculate the mass of the object

Solve the equation for the mass: \(m = \frac{9.067 \times 10^{-8}}{9.81}\) \(m \approx 9.244 \times 10^{-9} \; kg\) The mass of the object for which the gravitational force is equal to the electric force between the two protons is approximately \(9.244 \times 10^{-9} \; kg\).

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Most popular questions from this chapter

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