What must be a particle's charge-to-mass ratio if the magnitude of the gravitational force between two of these particles is equal to the magnitude of electric force between them?

To solve the problem, we need to equate the magnitudes of the gravitational force and the electric force between two identical particles. Here's the process:

### 1. Gravitational Force
The gravitational force between two particles of mass \( m \) separated by a distance \( r \) is given by Newton’s law of gravitation:
\[
F_g = \frac{G m^2}{r^2}
\]
where:
- \( F_g \) is the gravitational force,
- \( G \) is the gravitational constant (\( 6.674 \times 10^{-11} \, \text{Nm}^2/\text{kg}^2 \)),
- \( m \) is the mass of each particle,
- \( r \) is the distance between the particles.

### 2. Electric Force
The electric force between two particles with charge \( q \) separated by a distance \( r \) is given by Coulomb’s law:
\[
F_e = \frac{k_e q^2}{r^2}
\]
where:
- \( F_e \) is the electric force,
- \( k_e \) is Coulomb’s constant (\( 8.988 \times 10^9 \, \text{Nm}^2/\text{C}^2 \)),
- \( q \) is the charge of each particle,
- \( r \) is the distance between the particles.

### 3. Equating the Forces
We are told that the magnitudes of the gravitational and electric forces are equal. Therefore, we set the two forces equal to each other:
\[
\frac{G m^2}{r^2} = \frac{k_e q^2}{r^2}
\]
The distance \( r \) cancels out from both sides, leaving:
\[
G m^2 = k_e q^2
\]

### 4. Charge-to-Mass Ratio
We want the charge-to-mass ratio \( \frac{q}{m} \). First, solve for \( q^2/m^2 \) by dividing both sides by \( m^2 \):
\[
\frac{q^2}{m^2} = \frac{G}{k_e}
\]
Taking the square root of both sides:
\[
\frac{q}{m} = \sqrt{\frac{G}{k_e}}
\]

### 5. Substituting Values
Substitute the known values of \( G \) and \( k_e \):
\[
\frac{q}{m} = \sqrt{\frac{6.674 \times 10^{-11}}{8.988 \times 10^9}} \, \text{C/kg}
\]

Let’s compute this.

The required charge-to-mass ratio \( \frac{q}{m} \) is approximately \( 8.62 \times 10^{-11} \, \text{C/kg} \).

Would you like more details or have any further questions?

Here are five related questions for further exploration:
1. How does the distance between two particles affect the ratio of gravitational to electric forces?
2. What happens if the particles have different charges but the same mass?
3. Can a particle's charge-to-mass ratio affect its behavior in electric and gravitational fields differently?
4. What would be the charge-to-mass ratio if the electric force were stronger than the gravitational force by a factor of 100?
5. How does this charge-to-mass ratio compare to that of common particles like electrons or protons?

**Tip:** The gravitational force is typically much weaker than the electric force, which is why objects like electrons are primarily affected by electromagnetic rather than gravitational interactions at small scales.

Calculate the charge-to-mass ratio of a particle when the gravitational and electric forces between two particles are equal. This physics problem involves applying Newton's law of gravitation and Coulomb's law. Suitable for undergraduate physics students.

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