Calculating Distance In An Electric Field A Charge Of 7.2 X 10^-5 C In A 4.8 X 10^5 N/C Field

In the realm of electromagnetism, understanding the relationships between charge, electric fields, and potential energy is crucial. This article delves into a problem involving a charge placed in an electric field, aiming to calculate the distance between the charge and the source of the electric field. We will explore the fundamental concepts and formulas required to solve this problem, providing a step-by-step explanation to enhance understanding.

Consider a charge of 7.2 x 10^-5 C positioned within an electric field characterized by a strength of 4.8 x 10^5 N/C. The electric potential energy of this charge is measured to be 75 J. The core objective is to determine the distance separating the charge from the origin of the electric field. This problem elegantly combines the concepts of electric fields, potential energy, and the relationship between them, offering a practical application of electromagnetism principles. Solving this problem requires a solid understanding of the fundamental equations governing electric forces and energy, as well as the ability to apply these equations in a systematic manner.

To tackle this problem effectively, it's essential to grasp the core concepts and formulas that govern the interaction between charges and electric fields. The electric force (F) exerted on a charge (q) in an electric field (E) is given by:

F = qE

This equation states that the force on a charge is directly proportional to both the magnitude of the charge and the strength of the electric field. The direction of the force depends on the sign of the charge; positive charges experience a force in the direction of the electric field, while negative charges experience a force in the opposite direction.

The electric potential energy (U) of a charge in an electric field is related to the work done to move the charge within the field. The change in potential energy (ΔU) when moving a charge q through a distance d in an electric field E is:

ΔU = -qEd

This equation highlights that the change in potential energy is negative if the work is done by the electric field and positive if work is done against the electric field. In the context of this problem, we are given the electric potential energy (U) and need to find the distance (d). Rearranging the formula, we get:

d = -U / (qE)

This rearranged formula is the key to solving the problem, as it directly relates the distance to the given quantities: potential energy, charge, and electric field strength. By understanding and applying these formulas, we can systematically approach the problem and arrive at the correct solution.

Now, let's apply these concepts to solve the problem step by step.

Step 1: Identify the Given Values

First, we need to identify the values provided in the problem statement:

• Charge (q) = 7.2 x 10^-5 C
• Electric Field Strength (E) = 4.8 x 10^5 N/C
• Electric Potential Energy (U) = 75 J

Step 2: Apply the Formula

We use the formula derived earlier:

d = -U / (qE)

Step 3: Substitute the Values

Substitute the given values into the formula:

d = -75 J / (7.2 x 10^-5 C * 4.8 x 10^5 N/C)

Step 4: Calculate the Distance

Perform the calculation:

d = -75 / (7.2 x 10^-5 * 4.8 x 10^5) d = -75 / 34.56 d ≈ -2.17 meters

Step 5: Interpret the Result

The negative sign indicates that the displacement is in the opposite direction of the electric force if we consider the reference point where the potential energy is zero. However, the distance itself is a magnitude and should be positive. Therefore, the distance between the charge and the source of the electric field is approximately 2.17 meters. This step is crucial in ensuring that the numerical result is interpreted correctly in the physical context of the problem.

In conclusion, by applying the principles of electromagnetism and the relevant formulas, we have successfully calculated the distance between a charge and the source of an electric field. The key to solving this problem lies in understanding the relationship between electric force, electric field strength, and electric potential energy. By systematically applying the formula:

d = -U / (qE)

we found the distance to be approximately 2.17 meters. This exercise demonstrates the practical application of physics concepts in determining spatial relationships within electric fields. The negative sign in the initial calculation indicates the direction relative to the electric force, but the magnitude of the distance is the absolute value. Understanding such nuances is critical in physics problem-solving. This method can be applied to various similar problems involving electric fields and potential energy, making it a valuable tool for students and professionals in physics and related fields.

To further enhance understanding, let's delve into additional insights related to this problem. The concept of electric potential energy is closely linked to the work done by or against the electric field. When a positive charge moves in the direction of an electric field, it loses potential energy, and the field does positive work on it. Conversely, if the charge moves against the field, it gains potential energy, and work is done against the field. This is analogous to gravitational potential energy, where an object loses potential energy as it falls and gains it as it is lifted.

The electric field itself is a vector field, meaning it has both magnitude and direction at every point in space. The strength of the electric field is a measure of the force it would exert on a unit positive charge placed at that point. Electric fields are created by electric charges; positive charges create fields that point away from them, while negative charges create fields that point towards them. The superposition principle applies to electric fields, meaning the total electric field at a point due to multiple charges is the vector sum of the individual fields created by each charge.

In more complex scenarios, the electric field might not be uniform, and the potential energy calculation could involve integration. For example, if the electric field varies with distance, the formula ΔU = -qEd is not directly applicable, and one must integrate the force over the distance to find the change in potential energy. Understanding these nuances is essential for tackling advanced problems in electromagnetism. Furthermore, the concept of electric potential, which is electric potential energy per unit charge, provides another useful perspective. Electric potential is a scalar quantity, making it easier to work with in some situations compared to the vector electric field. Lines of equipotential, which are surfaces where the electric potential is constant, are always perpendicular to the electric field lines, providing a visual and conceptual tool for analyzing electric fields.

The principles discussed in this article have numerous applications in real-world scenarios and technological devices. For instance, understanding electric fields and potential energy is crucial in the design and operation of capacitors, which are fundamental components in electronic circuits. Capacitors store electrical energy by accumulating charge on conductive plates separated by an insulator. The amount of energy stored in a capacitor depends on the capacitance, the voltage across the capacitor, and the electric field between the plates.

Another significant application is in particle accelerators, which use electric fields to accelerate charged particles to high speeds. These accelerated particles are used in scientific research to probe the fundamental structure of matter. The design of particle accelerators requires precise control over electric and magnetic fields to guide and accelerate the particles efficiently. Medical devices such as MRI machines and X-ray machines also rely on the principles of electromagnetism. MRI machines use strong magnetic fields and radio waves to create detailed images of the human body, while X-ray machines use high-energy photons to penetrate tissues and produce images of bones and organs.

Electrostatic painting is another practical application, where electrically charged paint particles are attracted to a grounded object, resulting in a uniform coating. This technique is widely used in the automotive industry and other manufacturing processes. Furthermore, understanding electric fields is essential in designing lightning protection systems. Lightning rods provide a preferred path for lightning to discharge, protecting buildings and other structures from damage. These examples illustrate the broad applicability of the concepts discussed in this article and underscore the importance of a solid understanding of electromagnetism in various fields of science and technology.

This comprehensive exploration has provided a detailed understanding of how to calculate the distance between a charge and the source of an electric field, given the charge's electric potential energy and the field strength. We revisited the core formulas, including F = qE and ΔU = -qEd, and demonstrated their application in solving the problem step by step. By identifying the known values, substituting them into the formula, and interpreting the result, we found the distance to be approximately 2.17 meters. Moreover, we expanded on the fundamental concepts, discussing the relationship between potential energy and work, the nature of electric fields, and the importance of understanding vector and scalar quantities in electromagnetism. The additional insights provided a deeper appreciation of the problem's context and its connections to broader concepts in physics.

Furthermore, we highlighted the wide-ranging applications of these principles in real-world scenarios and technological devices, from capacitors and particle accelerators to medical imaging and electrostatic painting. These examples underscore the practical relevance of electromagnetism and its significance in various fields of science and engineering. Understanding these concepts is not only essential for academic pursuits but also for contributing to technological advancements and solving real-world problems. This article serves as a valuable resource for students, educators, and anyone interested in gaining a deeper understanding of electric fields and potential energy. By mastering these fundamental principles, one can unlock a greater appreciation for the physical world and the technologies that shape our lives.