Force between a System of Discrete Particles

In physics, understanding the force between a system of discrete particles is crucial for analyzing the interactions and dynamics of particle systems. These forces can be gravitational, electrostatic, magnetic, or any other fundamental force depending on the context. Here, we will focus on gravitational and electrostatic forces as examples.

Gravitational Force

Newton's Law of Universal Gravitation states that every particle in the universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.

The gravitational force between two particles with masses $m_1$ and $m_2$ separated by a distance $r$ is given by:

$$ F_g = G \frac{m_1 m_2}{r^2} $$

where $G$ is the gravitational constant.

Electrostatic Force

Coulomb's Law describes the electrostatic force between two charged particles. The force is directly proportional to the product of the charges and inversely proportional to the square of the distance between them.

The electrostatic force between two charges $q_1$ and $q_2$ separated by a distance $r$ is given by:

$$ F_e = k \frac{q_1 q_2}{r^2} $$

where $k$ is Coulomb's constant.

Differences and Important Points

Aspect	Gravitational Force	Electrostatic Force
Fundamental Quantity	Mass	Charge
Force Law	Newton's Law	Coulomb's Law
Proportionality	Directly proportional to the product of masses	Directly proportional to the product of charges
Inverse Square Law	Inversely proportional to the square of distance	Inversely proportional to the square of distance
Constant	Gravitational constant ($G$)	Coulomb's constant ($k$)
Nature of Force	Always attractive	Can be attractive or repulsive
Relative Strength	Weaker force	Stronger force

Examples

Example 1: Gravitational Force between Two Particles

Consider two particles with masses $m_1 = 5 \, \text{kg}$ and $m_2 = 10 \, \text{kg}$ separated by a distance of $r = 2 \, \text{m}$. The gravitational force between them is calculated as:

$$ F_g = G \frac{m_1 m_2}{r^2} = 6.674 \times 10^{-11} \, \text{Nm}^2/\text{kg}^2 \times \frac{5 \, \text{kg} \times 10 \, \text{kg}}{(2 \, \text{m})^2} = 8.34 \times 10^{-10} \, \text{N} $$

Example 2: Electrostatic Force between Two Charges

Consider two charges $q_1 = 1 \times 10^{-6} \, \text{C}$ and $q_2 = -2 \times 10^{-6} \, \text{C}$ separated by a distance of $r = 0.5 \, \text{m}$. The electrostatic force between them is calculated as:

$$ F_e = k \frac{q_1 q_2}{r^2} = 8.987 \times 10^9 \, \text{Nm}^2/\text{C}^2 \times \frac{1 \times 10^{-6} \, \text{C} \times -2 \times 10^{-6} \, \text{C}}{(0.5 \, \text{m})^2} = -7.19 \times 10^{-2} \, \text{N} $$

The negative sign indicates that the force is attractive.

Conclusion

The force between a system of discrete particles is a fundamental concept in physics that allows us to understand and predict the behavior of particle systems. Whether dealing with celestial bodies or subatomic particles, the principles of gravitational and electrostatic forces play a crucial role in the interactions within the universe.