### Physics Force Between Multiple Charges

##### Calculation of Magnitude of Force Between Two Charges

##### Coulomb’s Law

The magnitude of the electrostatic force between two point charges is given by Coulomb’s law:

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

where:

- $F$ is the magnitude of the force in newtons (N)
- $k$ is the electrostatic constant, approximately $8.988 × 10^9$ N m²/C²
- $q_1$ and $q_2$ are the magnitudes of the charges in coulombs (C)
- $r$ is the distance between the charges in meters (m)

##### Steps to Calculate the Magnitude of Force Between Two Charges

- Identify the two charges and their magnitudes.
- Determine the distance between the charges.
- Substitute the values of $q_1$, $q_2$, and $r$ into Coulomb’s law to calculate the magnitude of the force.

##### Example

Calculate the magnitude of the electrostatic force between two charges of $3\times10^{-6}$ C and $-2\times10^{-6}$ C separated by a distance of $0.5$ m.

**Solution:**

- The magnitudes of the charges are $q_1 = 3\times10^{-6}$ C and $q_2 = 2\times10^{-6}$ C.
- The distance between the charges is $r = 0.5$ m.
- Substituting these values into Coulomb’s law, we get:

$$F = k\frac{|q_1 q_2|}{r^2} = (8.988 × 10^9\text{ N m}^2/\text{C}^2)\frac{(3\times10^{-6}\text{ C})(2\times10^{-6}\text{ C})}{(0.5\text{ m})^2}$$

$$F = 5.39 × 10^{-3}\text{ N}$$

Therefore, the magnitude of the electrostatic force between the two charges is $5.39 × 10^{-3}$ N.

##### Derivation for Force Acting Between Multiple Charges

Coulomb’s law states that the force between two point charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. The force is also directed along the line connecting the two charges.

The mathematical expression for Coulomb’s law is:

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

Where:

- $F$ is the force between the two charges in newtons (N)
- $k$ is Coulomb’s constant, which is approximately $8.988 \times 10^9$ $N m^2/C^2$
- $q_1$ and $q_2$ are the magnitudes of the two charges in coulombs (C)
- $r$ is the distance between the two charges in meters (m)

##### Force Between Multiple Charges

The force between multiple charges can be calculated by using the principle of superposition. This principle states that the net force on a charge due to multiple other charges is equal to the vector sum of the forces due to each individual charge.

To calculate the force between multiple charges, we can first calculate the force between each pair of charges using Coulomb’s law. Then, we can add up these forces vectorially to get the net force.

For example, consider three charges $q_1$, $q_2$, and $q_3$ located at positions $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$, respectively. The force on charge $q_1$ due to charge $q_2$ is given by:

$$F_{12} = k\frac{q_1 q_2}{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

The force on charge $q_1$ due to charge $q_3$ is given by:

$$F_{13} = k\frac{q_1 q_3}{(x_3 - x_1)^2 + (y_3 - y_1)^2}$$

The net force on charge $q_1$ is then given by:

$$F_1 = F_{12} + F_{13}$$

We can calculate the forces on charges $q_2$ and $q_3$ in a similar manner.

##### Example

Consider three charges $q_1 = 1 \mu C$, $q_2 = 2 \mu C$, and $q_3 = 3 \mu C$ located at positions $(0, 0)$, $(1, 0)$, and $(0, 1)$ meters, respectively. The force on charge $q_1$ due to charge $q_2$ is given by:

$$F_{12} = k\frac{q_1 q_2}{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

$$F_{12} = (8.988 \times 10^9 \text{ N m}^2/\text{C}^2)\frac{(1 \times 10^{-6} \text{ C})(2 \times 10^{-6} \text{ C})}{(1 - 0)^2 + (0 - 0)^2}$$

$$F_{12} = 17.976 \times 10^{-3} \text{ N}$$

The force on charge $q_1$ due to charge $q_3$ is given by:

$$F_{13} = k\frac{q_1 q_3}{(x_3 - x_1)^2 + (y_3 - y_1)^2}$$

$$F_{13} = (8.988 \times 10^9 \text{ N m}^2/\text{C}^2)\frac{(1 \times 10^{-6} \text{ C})(3 \times 10^{-6} \text{ C})}{(0 - 0)^2 + (1 - 0)^2}$$

$$F_{13} = 26.964 \times 10^{-3} \text{ N}$$

The net force on charge $q_1$ is then given by:

$$F_1 = F_{12} + F_{13}$$

$$F_1 = 17.976 \times 10^{-3} \text{ N} + 26.964 \times 10^{-3} \text{ N}$$

$$F_1 = 44.94 \times 10^{-3} \text{ N}$$

The force on charge $q_2$ due to charge $q_1$ is equal in magnitude but opposite in direction to the force on charge $q_1$ due to charge $q_2$. The force on charge $q_3$ due to charge $q_1$ is also equal in magnitude but opposite in direction to the force on charge $q_1$ due to charge $q_3$.

##### Solved Examples on Force Between Multiple Charges

In electrostatics, the force between two point charges is given by Coulomb’s law:

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

where:

- $F$ is the force between the two charges in newtons (N)
- $k$ is Coulomb’s constant $(\approx 8.99 \times 10^9 \text{ N}\cdot\text{m}^2/\text{C}^2)$
- $q_1$ and $q_2$ are the magnitudes of the two charges in coulombs (C)
- $r$ is the distance between the two charges in meters (m)

The force between multiple charges can be found by using the principle of superposition. This principle states that the net force on a charge due to multiple other charges is the vector sum of the forces due to each individual charge.

##### Example 1: Force Between Three Charges

Consider three point charges $q_1 = 1 \mu \text{C}$, $q_2 = 2 \mu \text{C}$, and $q_3 = 3 \mu \text{C}$ located at the corners of an equilateral triangle with side length $a = 1 \text{ m}$. Find the net force on charge $q_1$.

**Solution:**

The distance between each pair of charges is:

$$r = \sqrt{a^2 + a^2} = \sqrt{2} \text{ m}$$

The force on charge $q_1$ due to charge $q_2$ is:

$$F_{12} = k\frac{q_1 q_2}{r^2} = (8.99 \times 10^9 \text{ N}\cdot\text{m}^2/\text{C}^2)\frac{(1 \times 10^{-6} \text{ C})(2 \times 10^{-6} \text{ C})}{(\sqrt{2} \text{ m})^2}$$

$$F_{12} = 5.06 \times 10^{-3} \text{ N}$$

The force on charge $q_1$ due to charge $q_3$ is:

$$F_{13} = k\frac{q_1 q_3}{r^2} = (8.99 \times 10^9 \text{ N}\cdot\text{m}^2/\text{C}^2)\frac{(1 \times 10^{-6} \text{ C})(3 \times 10^{-6} \text{ C})}{(\sqrt{2} \text{ m})^2}$$

$$F_{13} = 7.59 \times 10^{-3} \text{ N}$$

The net force on charge $q_1$ is:

$$F_{net} = F_{12} + F_{13} = 5.06 \times 10^{-3} \text{ N} + 7.59 \times 10^{-3} \text{ N}$$

$$F_{net} = 1.27 \times 10^{-2} \text{ N}$$

The net force on charge $q_1$ is $1.27 \times 10^{-2} \text{ N}$ directed at an angle of $30^\circ$ above the horizontal.

##### Example 2: Force on a Charge in an Electric Field

Consider a point charge $q = 1 \mu \text{C}$ located in an electric field $\overrightarrow{E} = 1000 \text{ N/C}$ directed to the right. Find the force on the charge.

**Solution:**

The force on the charge is given by:

$$\overrightarrow{F} = q\overrightarrow{E}$$

$$F = qE = (1 \times 10^{-6} \text{ C})(1000 \text{ N/C})$$

$$F = 1 \times 10^{-3} \text{ N}$$

The force on the charge is $1 \times 10^{-3} \text{ N}$ directed to the right.

##### Force Between Multiple Charges FAQs

##### What is the force between multiple charges?

The force between multiple charges is the vector sum of the forces between each pair of charges. The force between two charges is given by Coulomb’s law:

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

where:

- $F$ is the force in newtons (N)
- $k$ is Coulomb’s constant $(\approx 8.99 \times 10^9 \text{ N}\cdot\text{m}^2/\text{C}^2)$
- $q_1$ and $q_2$ are the magnitudes of the charges in coulombs (C)
- $r$ is the distance between the charges in meters (m)

##### What is the direction of the force between multiple charges?

The direction of the force between multiple charges is the same as the direction of the net force between the charges. The net force is the vector sum of the forces between each pair of charges.

##### What is the magnitude of the force between multiple charges?

The magnitude of the force between multiple charges is the square root of the sum of the squares of the magnitudes of the forces between each pair of charges.

##### How do you calculate the force between multiple charges?

To calculate the force between multiple charges, you must first calculate the force between each pair of charges. Then, you must add the forces together to find the net force.

##### What are some examples of the force between multiple charges?

Some examples of the force between multiple charges include:

- The force between two protons in a nucleus
- The force between two electrons in an atom
- The force between two ions in a solution
- The force between two charged particles in a plasma

##### What are the applications of the force between multiple charges?

The force between multiple charges has many applications, including:

- Understanding the structure of atoms and molecules
- Understanding the behavior of plasmas
- Designing particle accelerators
- Developing new materials

##### Conclusion

The force between multiple charges is a fundamental concept in physics. It is used to understand a wide variety of phenomena, from the structure of atoms to the behavior of plasmas.