The Force Between Two Identical Charges Calculated Using Coulomb’s Law

A common problem in physics and electrical engineering involves determining the magnitude of identical charges given the force between them. This article will walk you through the process of solving such a problem using Coulomb's Law. We will start by understanding the given scenario and then proceed to apply the formula to find the charges.

Understanding the Problem

Given that the force between two identical charges separated by 1 cm is 90 N, the objective is to find the magnitude of these charges. This can be done using Coulomb's Law, which defines the electrostatic force between two point charges.

Applying Coulomb's Law

According to Coulomb's Law, the force (F) between two point charges is given by:

[ F kfrac{q_1 q_2}{r^2} ]

where:

F is the magnitude of the force between the charges (90 N in this case) k is Coulomb's constant ((8.99 times 10^9 , text{Nm}^2/text{C}^2)) q1 and q2 are the magnitudes of the charges. Since the charges are identical, we can denote them both as (q). r is the distance between the charges (1 cm 0.01 m).

Substituting these values into the formula, we get:

[ F kfrac{q^2}{r^2} ]

Since (r^2 0.01 , text{m}^2), the equation simplifies to:

[ 90 8.99 times 10^9 times frac{q^2}{0.0001} ]

Further simplifying the equation:

[ q^2 90 times 0.0001 times frac{1}{8.99 times 10^9} ]

Performing the calculation:

[ q^2 90 times 10^{-4} times 10^{-9} times frac{1}{8.99} ]

This results in:

[ q^2 approx 1.00011 times 10^{-12} ]

Taking the square root of both sides:

[ q approx sqrt{1.00011 times 10^{-12}} ]

Therefore:

[ q approx 1.000055 times 10^{-6} , text{C} ]

Thus, the magnitude of each charge is approximately:

[ q approx 1.00 , mu text{C} ]

Alternative Method: Another Way to Solve

For another perspective, we can revisit the formula and apply it differently:

[ F k frac{q^2}{r^2} 90 , text{N} , 1 , text{cm}^{-2} 1 times 10^{-4} , text{m}^2 ]

Rearranging the formula to solve for (q):

[ q sqrt{frac{F r^2}{k}} ]

Substituting the values:

[ q sqrt{frac{90 times 10^{-4}}{9 times 10^9}} ]

Further calculations:

[ q sqrt{1.00011 times 10^{-12}} ]

Thus, the magnitude of each charge is approximately:

[ q approx 10^{-6} , text{C} ]

This confirms that the magnitude of the two identical charges is approximately (1.00 mu text{C}) each.

Conclusion

Coupling the principles of physics with mathematical calculations, we successfully found the magnitude of the charges. Using both Coulomb's Law and the rearranged formula, we arrived at a consistent result, validating the accuracy of the solution.

Additional Reading

For further exploration into this topic, students and professionals can refer to textbooks on classical mechanics and electrostatics. Understanding Coulomb’s Law is fundamental to the study of electricity and magnetism.