Calculating Electric Field Strength 0.020 M From A 12 ΜC Charge

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In the realm of electromagnetism, understanding the electric field is crucial. The electric field is a vector field that describes the force exerted on an electric charge at any point in space. It's a fundamental concept in physics, bridging the gap between electric charges and the forces they exert on each other. This article delves into calculating the electric field strength at a specific distance from a point charge, providing a comprehensive explanation and a step-by-step solution to a common problem in electrostatics.

The electric field strength is a measure of the force that would be exerted on a positive test charge placed at a given point in the field. It is a vector quantity, meaning it has both magnitude and direction. The magnitude of the electric field strength, often denoted by E, is defined as the force per unit charge. Mathematically, this is expressed as:

E = F / q

where:

  • E is the electric field strength (measured in Newtons per Coulomb, N/C)
  • F is the electric force (measured in Newtons, N)
  • q is the test charge (measured in Coulombs, C)

For a point charge, the electric field strength can be calculated using Coulomb's law. Coulomb's law describes 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. The formula for the electric field strength due to a point charge Q at a distance r is given by:

E = k |Q| / r^2

where:

  • E is the electric field strength (N/C)
  • k is Coulomb's constant (approximately 8.99 × 10^9 N⋅m²/C²)
  • |Q| is the absolute value of the charge (C)
  • r is the distance from the charge (m)

The direction of the electric field is radially outward from a positive charge and radially inward towards a negative charge. This means that if you place a positive test charge in the electric field of a positive source charge, the test charge will experience a repulsive force, pushing it away from the source charge. Conversely, if the source charge is negative, the positive test charge will experience an attractive force, pulling it towards the source charge.

Let's tackle a specific problem to illustrate the application of the electric field strength formula. The problem asks: What is the strength of the electric field 0.020 meters away from a 12 microcoulomb (12 μC) charge? We are also given the value of Coulomb's constant, k = 8.99 × 10^9 N⋅m²/C². This is a classic problem in electrostatics, requiring us to calculate the magnitude of the electric field at a given distance from a point charge.

This problem provides a practical scenario for understanding the concept of electric field strength. It involves a point charge, which is a simplified model of a charged object where the charge is considered to be concentrated at a single point in space. This approximation is valid when the dimensions of the charged object are much smaller than the distance at which the electric field is being calculated. The problem also gives us the distance from the charge, which is a crucial parameter in determining the electric field strength. The larger the distance, the weaker the electric field, as the force between charges diminishes with increasing separation.

The given charge of 12 μC (microcoulombs) is a relatively small amount of charge, but it's still significant enough to create a measurable electric field at a distance of 0.020 meters. The prefix "micro" (μ) represents 10^-6, so 12 μC is equal to 12 × 10^-6 Coulombs. This conversion is essential for using the correct units in the formula. Coulomb's constant (k) is a fundamental constant in electrostatics, representing the proportionality factor in Coulomb's law. Its value is approximately 8.99 × 10^9 N⋅m²/C², and it plays a critical role in determining the strength of the electrostatic force and, consequently, the electric field strength.

Now, let's break down the solution step-by-step:

Step 1: Identify the Known Values

First, we need to identify the known quantities from the problem statement:

  • Charge (Q) = 12 μC = 12 × 10^-6 C
  • Distance (r) = 0.020 m
  • Coulomb's constant (k) = 8.99 × 10^9 N⋅m²/C²

Step 2: Apply the Formula

Next, we apply the formula for the electric field strength due to a point charge:

E = k |Q| / r^2

Step 3: Substitute the Values

Substitute the known values into the formula:

E = (8.99 × 10^9 N⋅m²/C²) × |12 × 10^-6 C| / (0.020 m)^2

Step 4: Calculate the Electric Field Strength

Now, perform the calculation:

E = (8.99 × 10^9 N⋅m²/C²) × (12 × 10^-6 C) / (0.0004 m²) E = (107.88 × 10^3 N⋅m²/C) / (0.0004 m²) E = 269.7 × 10^6 N/C E = 2.70 × 10^7 N/C (approximately)

Therefore, the strength of the electric field 0.020 meters from a 12 μC charge is approximately 2.70 × 10^7 N/C.

The calculated electric field strength of 2.70 × 10^7 N/C is a significant value. This indicates a strong electric field at this distance from the charge. The electric field strength is directly proportional to the magnitude of the charge and inversely proportional to the square of the distance. This means that as you move further away from the charge, the electric field strength decreases rapidly.

The direction of the electric field at this point is radially outward from the charge, since the charge is positive. This means that if a positive test charge were placed at this point, it would experience a force pushing it away from the 12 μC charge.

The magnitude of the electric field provides insight into the force a charged particle would experience. For instance, if we placed a proton (charge = 1.602 × 10^-19 C) at this location, the force on the proton could be calculated using the formula F = qE. This would give us:

F = (1.602 × 10^-19 C) × (2.70 × 10^7 N/C) ≈ 4.33 × 10^-12 N

This force, although seemingly small, is significant at the atomic and subatomic levels, where electrostatic forces play a crucial role in the interactions between particles.

Several factors influence the strength of the electric field. Understanding these factors is key to grasping the behavior of electric fields in various scenarios.

1. Magnitude of the Charge

The electric field strength is directly proportional to the magnitude of the charge creating the field. This means that a larger charge will produce a stronger electric field at a given distance. As seen in the formula E = k |Q| / r^2, the electric field (E) increases linearly with the absolute value of the charge (|Q|). If we double the charge, the electric field strength doubles, assuming the distance remains constant. This relationship underscores the fundamental connection between charge and the electric field it generates.

2. Distance from the Charge

Distance plays a critical role in determining the strength of the electric field. The electric field strength is inversely proportional to the square of the distance from the charge. This inverse square relationship means that as the distance (r) increases, the electric field strength (E) decreases dramatically. Doubling the distance reduces the electric field strength to one-fourth of its original value. This rapid decrease with distance is a characteristic feature of electric fields generated by point charges and is crucial in understanding how electric forces diminish over space.

3. Medium Surrounding the Charge

The medium surrounding the charge can also affect the electric field strength. In a vacuum, the electric field has its maximum strength. However, when a charge is placed in a material medium, the electric field is reduced due to the polarization of the medium. The molecules in the medium align themselves in response to the electric field, creating an opposing field that weakens the net electric field. This effect is quantified by the dielectric constant of the medium. The electric field strength in a medium is given by:

E = E₀ / εᵣ

where:

  • E₀ is the electric field strength in a vacuum
  • εᵣ is the relative permittivity (dielectric constant) of the medium

The dielectric constant is a dimensionless quantity that indicates how much the electric field is reduced in the medium compared to a vacuum. Materials with high dielectric constants, such as water, can significantly reduce the electric field strength.

4. Presence of Other Charges

The presence of other charges in the vicinity can also influence the electric field strength at a particular point. Electric fields, being vector fields, obey the principle of superposition. This principle states that the total electric field at a point due to multiple charges is the vector sum of the electric fields created by each individual charge at that point. The electric fields can either add up constructively, increasing the field strength, or cancel each other out partially or completely, depending on the magnitudes, signs, and positions of the charges.

The understanding of electric fields and their strengths is not just an academic exercise; it has numerous real-world applications that underpin many technologies and phenomena we encounter daily.

1. Electrostatic Precipitators

Electrostatic precipitators are used in industrial settings to remove particulate matter from exhaust gases, thereby reducing air pollution. These devices utilize strong electric fields to charge the particles in the gas stream. The charged particles are then attracted to oppositely charged plates, where they are collected and removed. The efficiency of these precipitators depends critically on the electric field strength; a stronger electric field leads to more effective particle charging and collection.

2. Medical Imaging (MRI)

Magnetic Resonance Imaging (MRI) relies on strong magnetic fields and radiofrequency electromagnetic fields to produce detailed images of the internal structures of the body. While MRI primarily uses magnetic fields, electric fields are also involved in the interaction of radiofrequency waves with the body's tissues. The precise control and understanding of these electric fields are crucial for obtaining high-quality images and ensuring patient safety.

3. Electronics and Semiconductors

Electric fields are fundamental to the operation of electronic devices, particularly in semiconductors. Transistors, the building blocks of modern electronics, use electric fields to control the flow of current. The strength and direction of the electric field in a transistor determine its switching behavior and amplification capabilities. Understanding electric fields is thus essential for designing and optimizing electronic circuits.

4. Lightning Protection

Lightning is a dramatic example of the power of electric fields in nature. Lightning rods are designed to protect buildings from lightning strikes by providing a preferential path for the electric discharge. The sharp point of the lightning rod concentrates the electric field, encouraging the lightning to strike there rather than the building itself. The rod then conducts the current safely to the ground, preventing damage to the structure. The effectiveness of lightning rods hinges on the principles of electric field concentration and conduction.

5. Particle Accelerators

Particle accelerators use strong electric fields to accelerate charged particles to very high speeds. These high-energy particles are used in fundamental research to probe the structure of matter and in medical applications such as cancer therapy. The strength and uniformity of the electric fields in these accelerators are crucial for achieving the desired particle energies and beam quality.

In summary, calculating electric field strength is a fundamental concept in electromagnetism with wide-ranging applications. By understanding the factors that influence electric field strength and the formula for its calculation, we can analyze and predict the behavior of electric fields in various scenarios. This knowledge is crucial for engineers, physicists, and anyone working with electrical phenomena. The example problem discussed in this article provides a clear illustration of how to apply the electric field strength formula, and the real-world applications highlight the importance of this concept in technology and beyond.