91Ó°ÊÓ

Coulomb's Law is essential for understanding how electric charges interact. It describes the force between two point charges. The formula is \[ F = \frac{k |q_1 q_2|}{r^2} \]where:- \(F\) is the force between the charges,- \(k\) is Coulomb's constant \( (8.99 \times 10^9 \, \text{N} \cdot \text{m}^2/\text{C}^2) \),- \(q_1\) and \(q_2\) are the magnitudes of the charges,- \(r\) is the distance between the charges.
In this exercise, we use Coulomb's law specifically to determine the electric fields resulting from point charges. The force changes depending on the sign of the charges:

• If both charges have the same sign, the force is repulsive, pushing the charges apart.
• If the charges have opposite signs, the force is attractive, pulling the charges together.

The law helps to understand the attraction and repulsion occurring between charges in space.

A point charge refers to a charge that is considered to be concentrated in a single point in space. It simplifies how we calculate electric fields and forces.
Point charges are theoretical constructs, useful for understanding electrical interactions without worrying about the actual size of the charge. In real life, this assumption might apply if:

• The distance between charges is large compared to their size.
• The spatial extent of the charge is negligible in analysis.

Here, the exercise involves two point charges \( q_1 = -4.00 \, \text{nC} \)and \( q_2 \), allowing us to apply formulas directly.

• \( q_1 \) is located at \( x = 0.60 \, \text{m} \) on the x-axis.
• \( q_2 \) is located at \( x = -1.20 \, \text{m} \) on the x-axis.

Understanding point charges lets us calculate electric fields efficiently, using theoretical point interactions, as shown in this exercise.

The direction of an electric field indicates the force experienced by a positive test charge placed in the field. Electric fields point away from positive charges and towards negative charges.
For example, in this exercise, the electric field produced by the negative point charge \( q_1 = -4.00 \, \text{nC} \) at the origin points toward \( q_1 \). Therefore, it is in the \(-x\)-direction. Knowing the direction helps predict the behaviour of charges:

• Fields from positive charges repel other positive charges and attract negative charges.
• Fields from negative charges attract positive charges and repel other negative charges.

In scenario (a), we need the net field to point in the \(+x\)-direction, so \( q_2 \) must create a stronger opposing field. In scenario (b), the field should be \(-x\), complementing the field of \( q_1 \). Understanding field direction is crucial to solve problems involving multiple charges.

The net electric field at a point is the vector sum of the electric fields due to individual charges at that point. Each charge produces its field, and these fields combine;\( E_{\text{net}} = E_1 + E_2 + \ldots \)For this exercise:

• The electric field at the origin due to \( q_1 \) is \(99.9 \, \text{N/C}\) in the \(-x\)-direction.
• Scenario (a) requires \( E_{\text{net}} = 50.0 \, \text{N/C} \) in the \(+x\)-direction.
• Scenario (b) needs \( E_{\text{net}} = 50.0 \, \text{N/C} \) in the \(-x\)-direction.

To achieve these net fields, we adjust \( q_2 \) appropriately, considering both its magnitude and sign to counter or complement the field from \( q_1 \). Calculating the net electric field helps in understanding charge interactions and predicting the electric field configuration at any point in space.