Calculating The Distance Between Two Charges Using Coulombs Law

by ADMIN 64 views

In the realm of physics, particularly within the study of electromagnetism, understanding the forces between charged objects is fundamental. Coulomb's Law provides a quantitative description of this interaction, stating that the force between two point charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. This article will delve into a practical application of Coulomb's Law, specifically calculating the distance between two charges given their magnitudes and the force of attraction between them. We will explore the underlying principles, the mathematical formulation of Coulomb's Law, and a step-by-step solution to a problem involving two charges of $+5 imes 10^{-7} C$ and $-2 imes 10^{-7} C$ attracting each other with a certain force.

Understanding Coulomb's Law

At the heart of electrostatics lies Coulomb's Law, a cornerstone principle that governs the interactions between electrically charged objects. This law, formulated by French physicist Charles-Augustin de Coulomb in the late 18th century, elegantly describes the force exerted between two point charges. It states that the force is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance separating them. Mathematically, this relationship is expressed as:

F = k rac{|q_1 q_2|}{r^2}

Where:

  • F represents the electrostatic force between the charges.
  • k is Coulomb's constant, approximately equal to $8.9875 imes 10^9 N m2/C2$.
  • q1 and q2 are the magnitudes of the two charges.
  • r is the distance between the charges.

The force described by Coulomb's Law is a vector quantity, possessing both magnitude and direction. The direction of the force is along the line joining the two charges. If the charges have the same sign (both positive or both negative), the force is repulsive, pushing the charges apart. Conversely, if the charges have opposite signs (one positive and one negative), the force is attractive, pulling the charges together.

Coulomb's Law is analogous to Newton's Law of Universal Gravitation, which describes the gravitational force between two masses. However, there are key differences. Gravitational force is always attractive, while electrostatic force can be either attractive or repulsive. Furthermore, the electrostatic force is significantly stronger than the gravitational force for comparable charges and masses.

Applying Coulomb's Law: A Step-by-Step Approach

To effectively apply Coulomb's Law to solve problems, a systematic approach is crucial. The following steps provide a clear framework for tackling such problems:

  1. Identify the Knowns: Begin by carefully identifying the given information in the problem. This typically includes the magnitudes of the charges (q1 and q2) and the force (F) between them. In some cases, the problem might provide the distance (r) and ask for the force, or vice versa. Ensure that all values are expressed in consistent units, preferably SI units (Coulombs for charge, meters for distance, and Newtons for force).

  2. Write Down Coulomb's Law: Write the formula for Coulomb's Law: $F = k rac{|q_1 q_2|}{r^2}$. This step helps to visualize the relationship between the variables and ensures that the correct formula is being used.

  3. Rearrange the Formula (if necessary): If the problem requires solving for a variable other than the force (e.g., the distance r), rearrange the formula algebraically to isolate the desired variable. For instance, to solve for r, the formula can be rearranged as:

    r = \sqrt{k rac{|q_1 q_2|}{F}}

  4. Substitute the Values: Substitute the known values into the rearranged formula. Ensure that the values are substituted with their correct units.

  5. Calculate the Result: Perform the calculation carefully, paying attention to the order of operations and the units. Use a calculator if necessary to ensure accuracy.

  6. State the Answer with Units: Express the final answer with the appropriate units. For example, if calculating the distance, the answer should be expressed in meters.

  7. Consider the Direction of the Force: Determine whether the force is attractive or repulsive based on the signs of the charges. If the charges have opposite signs, the force is attractive; if they have the same sign, the force is repulsive.

Problem Statement

Two charges, one of $+5 imes 10^{-7} C$ and the other of $-2 imes 10^{-7} C$, attract each other with a force of 0.09 N. How far apart are they?

Solution

1. Identify the Knowns:

  • Charge 1 (q1) = $+5 imes 10^{-7} C$
  • Charge 2 (q2) = $-2 imes 10^{-7} C$
  • Force (F) = 0.09 N
  • Coulomb's constant (k) = $8.9875 imes 10^9 N m2/C2$

2. Write Down Coulomb's Law:

F = k rac{|q_1 q_2|}{r^2}

3. Rearrange the Formula to Solve for Distance (r):

To find the distance r, we need to rearrange Coulomb's Law:

r^2 = k rac{|q_1 q_2|}{F}

r = \sqrt{k rac{|q_1 q_2|}{F}}

4. Substitute the Values:

Now, substitute the known values into the rearranged formula:

r = \sqrt{(8.9875 imes 10^9 N m^2/C^2) rac{|(5 imes 10^{-7} C) (-2 imes 10^{-7} C)|}{0.09 N}}

5. Calculate the Result:

r = \sqrt{(8.9875 imes 10^9 N m^2/C^2) rac{10 imes 10^{-14} C^2}{0.09 N}}

r = \sqrt{ rac{8.9875 imes 10^{-4} N m^2}{0.09 N}}

r=9.986imes10−3m2r = \sqrt{9.986 imes 10^{-3} m^2}

r≈0.0999mr \approx 0.0999 m

6. State the Answer with Units:

The distance between the charges is approximately 0.0999 meters, or about 10 centimeters.

7. Consider the Direction of the Force:

Since the charges have opposite signs (one positive and one negative), the force between them is attractive, as stated in the problem.

Conclusion

In conclusion, by applying Coulomb's Law and following a systematic approach, we have successfully calculated the distance between two charges given their magnitudes and the force of attraction between them. This problem illustrates the practical application of Coulomb's Law in determining the electrostatic interactions between charged objects. Understanding Coulomb's Law is crucial for comprehending a wide range of phenomena in electromagnetism, from the behavior of atoms and molecules to the operation of electrical devices. The step-by-step method outlined in this article provides a valuable framework for solving similar problems involving electrostatic forces.

Keywords

  • Coulomb's Law
  • Electrostatic force
  • Electric charges
  • Distance between charges
  • Force of attraction
  • Magnitude of charge
  • Coulomb's constant
  • Electromagnetism
  • Electrostatics
  • Point charges
  • Calculating distance
  • Physics problems
  • Solved example
  • Step-by-step solution
  • Charge interaction
  • Attractive force
  • Repulsive force
  • SI units
  • Problem-solving