Calculating the Net Electric Force on a Charge

Consider the following setup: a 300 μC charge placed at the origin, a 60 μC charge located 2 cm from the origin, and a -500 μC charge situated 6 cm from the origin. Our goal is to determine the net electric force acting on the 300 μC charge.

To solve this problem, we will use Coulomb's Law and the superposition principle, which are fundamental concepts in electrostatics. Coulomb's Law dictates the force between two charged particles, while the superposition principle allows us to find the net force by adding individual forces.

Understanding the Problem and Setting Up Variables

Let's label the charges for clarity:

Charge at the origin: 0 Charge at 2 cm from the origin: 2 Charge at 6 cm from the origin: 6

Applying Coulomb's Law

Coulomb's Law is expressed as:

[ F k frac{q_1 q_2}{r^2} ]

Where:

F is the force between the two charges, k is Coulomb's constant (approximately (9 times 10^9 , text{Nm}^2/text{C}^2)), q1 and q2 are the charges involved, and r is the distance between the charges.

Force Due to the 60 μC Charge

First, we calculate the force due to the 60 μC charge (charge 2) on the 300 μC charge (charge 0):

( F_{20} 9 times 10^9 frac{300 times 10^{-6} times 60 times 10^{-6}}{(0.02)^2} ) ( 4 times 10^5 , text{N} )

This is a repulsive force, directed in the negative x-direction. Hence,

( vec{F}_{20} -4 times 10^5 , text{N} hat{x} )

Force Due to the -500 μC Charge

Next, we calculate the force due to the -500 μC charge (charge 6) on the 300 μC charge (charge 0):

( F_{60} 9 times 10^9 frac{300 times 10^{-6} times (-500 times 10^{-6})}{(0.06)^2} ) ( approx 3.75 times 10^5 , text{N} )

This is an attractive force, directed in the positive x-direction. Hence,

( vec{F}_{60} 3.75 times 10^5 , text{N} hat{x} approx 4 times 10^5 , text{N} hat{x} )

Superposition of Forces

According to the superposition principle, we add the individual forces vectorially:

[ vec{F} vec{F}_{20} vec{F}_{60} (-4 times 10^5 , text{N}) hat{x} (4 times 10^5 , text{N}) hat{x} 0 hat{x} ]

This indicates that the net force acting on the 300 μC charge is zero.

Conclusion

The net electric force on the 300 μC charge is zero. This means the charges at 2 cm and 6 cm cancel each other's forces exactly, resulting in no net force on the central charge.

Key Takeaways:

Coulomb's Law is used to calculate the force between charges. The superposition principle allows us to find the net force by summing individual forces. Understanding these principles is crucial in tackling electrostatic problems effectively.