Determine The Intensity Of The Electric Field (E) At The Midpoint P Between Two Point Charges With Values Q1= 8 NC And Q2= -3 NC, Separated By A Distance Of 14 Cm. Also, Calculate The Force That Would Act On A 2 NC Charge If Placed At P.

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In the realm of electromagnetism, understanding the electric field intensity and the forces that arise between charged particles is fundamental. This article delves into a specific scenario involving two point charges, q1 and q2, separated by a certain distance. We aim to determine the electric field intensity (E) at the midpoint between these charges and subsequently calculate the force that would act upon a third charge placed at that midpoint. This exploration provides a practical application of Coulomb's law and the principles of superposition in electrostatics. We will walk through the concepts of electric fields, point charges, Coulomb's law, and how to apply these principles to solve a numerical problem. This discussion not only aids in grasping the theoretical underpinnings but also in enhancing problem-solving skills crucial for physics and engineering disciplines.

Consider two point charges, q1 = 8 nC (nanoCoulombs) and q2 = -3 nC, separated by a distance of 14 cm. Our objectives are twofold:

  1. Determine the electric field intensity (E) at point P, the midpoint between the two charges.
  2. Calculate the force that would act on a charge of 2 nC if placed at point P.

This problem encapsulates core concepts in electromagnetism, specifically the superposition of electric fields and the electrostatic force exerted on a charge within an electric field. Solving this requires a clear understanding of Coulomb's law and the principle of superposition. We will proceed by first defining the key concepts and then applying them to solve the problem step by step.

Key Concepts

Before diving into the calculations, let's clarify the key concepts involved:

  • Electric Field: An electric field is a region around a charged particle or object within which a force would be exerted on other charged particles or objects. It is a vector field, meaning it has both magnitude and direction at every point in space. The electric field intensity (E) at a point is defined as the force per unit charge that would be exerted on a positive test charge placed at that point. Mathematically, it is given by E = F/q, where F is the force and q is the charge.
  • Point Charge: A point charge is an idealized concept representing an electric charge located at a single point in space. While physical charges have a finite size, the point charge approximation is valid when the size of the charge-bearing object is much smaller than the distances involved in the problem.
  • Coulomb's Law: Coulomb's law quantifies the electrostatic force 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 between them. Mathematically, the magnitude of the force F between two point charges q1 and q2 separated by a distance r is given by F = k * |q1 * q2| / r^2, where k is Coulomb's constant (approximately 8.9875 × 10^9 N m2/C2).
  • Superposition Principle: The principle of superposition states that the total electric field at a point due to multiple charges is the vector sum of the electric fields produced by each charge individually. Similarly, the total force on a charge due to multiple other charges is the vector sum of the individual forces. This principle is crucial for solving problems involving multiple charges.

Step-by-Step Solution

1. Determine the Electric Field Intensity Due to Each Charge

First, we calculate the electric field intensity at point P due to each charge individually. Point P is the midpoint between q1 and q2, which are separated by 14 cm. Therefore, the distance from each charge to point P is half of the separation distance:

r = 14 cm / 2 = 7 cm = 0.07 m

The electric field intensity due to a point charge q at a distance r is given by:

E = k * |q| / r^2

Where k is Coulomb's constant (approximately 8.9875 × 10^9 N m2/C2). Let's calculate the electric field due to q1 (8 nC) at point P:

E1 = (8.9875 × 10^9 N m2/C2) * (8 × 10^-9 C) / (0.07 m)^2

E1 = (8.9875 × 8) / 0.0049 N/C

E1 ≈ 14685.7 N/C

Since q1 is positive, the electric field E1 at point P points away from q1, towards q2.

Next, we calculate the electric field due to q2 (-3 nC) at point P:

E2 = (8.9875 × 10^9 N m2/C2) * (3 × 10^-9 C) / (0.07 m)^2

E2 = (8.9875 × 3) / 0.0049 N/C

E2 ≈ 5502.0 N/C

Since q2 is negative, the electric field E2 at point P points towards q2, which is the same direction as E1.

2. Apply the Superposition Principle

The total electric field intensity at point P is the vector sum of the electric fields due to q1 and q2. Since E1 and E2 are in the same direction (towards q2), we can simply add their magnitudes:

E = E1 + E2

E ≈ 14685.7 N/C + 5502.0 N/C

E ≈ 20187.7 N/C

Thus, the total electric field intensity at point P is approximately 20187.7 N/C, directed towards q2.

3. Calculate the Force on a 2 nC Charge

Now, we calculate the force that would act on a charge q3 = 2 nC if placed at point P. The force on a charge q in an electric field E is given by:

F = q * E

In this case, q = 2 nC = 2 × 10^-9 C and E ≈ 20187.7 N/C. Therefore,

F = (2 × 10^-9 C) * (20187.7 N/C)

F ≈ 4.03754 × 10^-5 N

The force on the 2 nC charge at point P is approximately 4.03754 × 10^-5 N. Since the electric field is directed towards q2, and q3 is positive, the force on q3 is also directed towards q2.

Summary of Results

  1. The electric field intensity (E) at the midpoint P between the two charges is approximately 20187.7 N/C, directed towards the negative charge q2.
  2. The force that would act on a 2 nC charge placed at point P is approximately 4.03754 × 10^-5 N, also directed towards q2.

Discussion and Implications

This problem illustrates the fundamental principles of electrostatics, including Coulomb's law and the superposition principle. The electric field intensity at a point in space is a crucial concept for understanding how charges interact. The magnitude and direction of the electric field provide a map of the forces that would be exerted on other charges placed in that field. By calculating the electric field due to multiple charges and then using the superposition principle, we can determine the net effect of these charges at any point.

The calculation of the force on the 2 nC charge highlights the direct relationship between electric field and force. A charge placed in an electric field experiences a force proportional to the magnitude of the charge and the electric field intensity. This principle is the basis for many applications, including electrostatic devices, particle accelerators, and understanding the behavior of charged particles in electric fields.

Factors Affecting Electric Field Intensity

Several factors influence the electric field intensity at a given point:

  • Magnitude of Charges: The electric field intensity is directly proportional to the magnitude of the charges creating the field. Larger charges produce stronger electric fields.
  • Distance from Charges: The electric field intensity is inversely proportional to the square of the distance from the charges. As distance increases, the field strength decreases rapidly.
  • Arrangement of Charges: The spatial arrangement of charges significantly impacts the electric field. The superposition principle dictates that the total field is the vector sum of individual fields, so the relative positions and orientations of charges determine the net field.
  • Medium Surrounding Charges: The medium in which the charges are placed affects the electric field. The presence of a dielectric material reduces the electric field intensity compared to that in a vacuum, due to the polarization of the dielectric.

Practical Applications

The concepts and calculations discussed in this article have numerous practical applications in various fields:

  • Electrostatic Devices: Devices like electrostatic precipitators, used for removing particulate matter from exhaust gases, rely on the principles of electric fields and forces to charge and collect particles.
  • Particle Accelerators: Particle accelerators use electric fields to accelerate charged particles to high speeds for scientific research. Understanding the electric field distribution is crucial for designing and operating these accelerators.
  • Electronics: Electric fields play a critical role in the functioning of electronic components such as capacitors, transistors, and integrated circuits. The performance of these devices depends on the precise control and manipulation of electric fields.
  • Medical Imaging: Techniques like electrocardiography (ECG) and electroencephalography (EEG) rely on detecting and interpreting electric potentials generated by the human body. These potentials are related to the electric fields produced by the electrical activity of the heart and brain.

Conclusion

Determining the electric field intensity and the forces between charges is a fundamental aspect of electromagnetism. This article demonstrated a step-by-step approach to calculating the electric field at the midpoint between two point charges and the force on a third charge placed at that point. By applying Coulomb's law and the superposition principle, we were able to solve the problem and gain insights into the behavior of electric fields and forces. The concepts discussed are not only essential for academic understanding but also have significant practical implications in various technological applications. A thorough understanding of electrostatics is crucial for anyone studying physics, engineering, or related fields. Through problems like this, we reinforce the theoretical knowledge and enhance our ability to apply these concepts in practical scenarios.

The study of electric fields and forces is a cornerstone of electromagnetism, with far-reaching implications across science and technology. By mastering the fundamental principles and problem-solving techniques, we can unlock a deeper understanding of the physical world and contribute to advancements in various fields. This article serves as a guide to the practical application of these concepts, encouraging readers to explore further and delve into the fascinating realm of electromagnetics.