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Chapter 16: Problem 14

Two metal spheres separated by a distance much greater than either sphere's radius have equal mass \(m\) and equal electric charge \(q .\) What is the ratio of charge to mass \(q / m\) in C/kg if the electrical and gravitational forces balance?

Short Answer

Expert verified
The ratio of charge to mass (q/m) for the two metal spheres is approximately 2.45 x 10^{-12} C/kg, when the electrical and gravitational forces balance.

Step by step solution

01

Write the equation for the electrical force

According to Coulomb's law, the electrical force between two charged objects is given by the equation: $$F_e = k \cdot \frac{q^2}{r^2}$$ where \(F_e\) is the electrical force, \(k\) is Coulomb's constant \((8.99 \times 10^9 Nm^2/C^2)\), \(q\) is the charge of each sphere, and \(r\) is the distance between the two spheres.
02

Write the equation for the gravitational force

According to Newton's law of universal gravitation, the gravitational force between two objects with masses \(m_1\) and \(m_2\) and distance \(r\) between their centers is given by the equation: $$F_g = G \cdot \frac{m_1 m_2}{r^2}$$ where \(F_g\) is the gravitational force and \(G\) is the gravitational constant \((6.67 \times 10^{-11} Nm^2/kg^2)\). Since the spheres have equal masses, we can write this as: $$F_g = G \cdot \frac{m^2}{r^2}$$
03

Equate the magnitudes of the electrical force and gravitational force

Since the electrical and gravitational forces balance each other, their magnitudes are equal: $$F_e = F_g$$ Using the expressions of \(F_e\) and \(F_g\) from steps 1 and 2: $$k \cdot \frac{q^2}{r^2} = G \cdot \frac{m^2}{r^2}$$
04

Solve the equation for the ratio \(q/m\)

We want to find the ratio \(q/m\), so let's isolate that term from the previous equation. $$q^2 = \frac{G}{k} m^2$$ Take the square root of both sides and then divide by \(m\): $$\frac{q}{m} = \sqrt{\frac{G}{k}}$$
05

Calculate the value of the ratio \(q/m\)

Using the given values for \(G\) and \(k\), we can determine the value for the ratio \(\frac{q}{m}\): $$\frac{q}{m} = \sqrt{\frac{6.67 \times 10^{-11} Nm^2/kg^2}{8.99 \times 10^9 Nm^2/C^2}}$$ $$\frac{q}{m} \approx 2.45 \times 10^{-12} \, C/kg$$ The ratio of charge to mass \(q/m\) for the two metal spheres is approximately \(2.45 \times 10^{-12}\) C/kg if the electrical and gravitational forces balance.

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