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Electric Field From A Line Of Charge

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Understanding the Electric Field from a Line of Charge



Electricity, a fundamental force of nature, governs much of our daily lives. Understanding electric fields is crucial to grasping how this force works. While a single point charge produces a readily calculable electric field, many real-world scenarios involve distributions of charge, such as a charged rod or a wire. This article focuses on a simplified yet important case: the electric field generated by an infinitely long line of charge. This seemingly abstract concept has significant practical applications in understanding phenomena like the behaviour of charged wires in electrical circuits.

1. What is an Electric Field?



Before diving into line charges, let's refresh our understanding of electric fields. An electric field is a region of space around a charged object where a force acts on another charged object. Think of it as an invisible influence that extends outward from the charge. This force is proportional to the magnitude of the charges involved and inversely proportional to the square of the distance between them (Coulomb's Law). Electric field lines visually represent this field, pointing away from positive charges and towards negative charges. The denser the lines, the stronger the field at that point.


2. The Infinite Line of Charge: A Simplification



Calculating the electric field from a realistically finite line of charge involves complex integration. To simplify the problem, we often consider an infinitely long line of charge. While not entirely realistic, this approximation provides a good model for situations where the length of the charged object is significantly larger than the distance at which we're measuring the field. This simplification allows us to use symmetry to our advantage, making the calculation manageable.

3. Calculating the Electric Field: Using Gauss's Law



The most efficient way to determine the electric field of an infinitely long line of charge is using Gauss's Law. This law states that the total electric flux through a closed surface is proportional to the enclosed charge. For our line charge, we strategically choose a cylindrical Gaussian surface.

Here's a breakdown:

Gaussian Surface: We imagine a cylinder surrounding a section of the infinitely long line of charge. The cylinder's axis is aligned with the line charge. The electric field will be perpendicular to the curved surface of the cylinder and parallel to the flat ends.

Symmetry: Due to the symmetry of the infinite line charge, the electric field magnitude is constant across the curved surface of the cylinder.

Gauss's Law Application: Gauss's Law allows us to relate the electric field (E) on the cylindrical surface, the length of the cylinder (L), and the linear charge density (λ – charge per unit length) of the line:

2πrLE = λL/ε₀

where ε₀ is the permittivity of free space (a constant).

Solving for E: Solving for E, we obtain:

E = λ / (2πε₀r)

This equation shows that the electric field strength is directly proportional to the linear charge density (λ) and inversely proportional to the distance (r) from the line charge. The field strength decreases as we move further away from the line.


4. Practical Examples and Applications



Consider a high-voltage transmission line. While not infinitely long, it's long enough that the infinite line approximation provides a reasonably accurate estimate of the electric field in its vicinity. This helps engineers design safety measures to prevent electrical hazards. Similarly, the concept is crucial in understanding the fields around coaxial cables used in electronics, where a central conductor is surrounded by a cylindrical shield.


5. Key Takeaways and Insights



The electric field from an infinitely long line of charge is radial, pointing directly away (or towards, depending on the charge's sign) from the line.
Its magnitude is inversely proportional to the distance from the line, meaning it weakens with distance.
Gauss's Law significantly simplifies the calculation of this field compared to using Coulomb's Law directly.
This simplified model provides a useful approximation for many real-world scenarios involving long, thin charged objects.


FAQs



1. Why do we use an infinitely long line of charge? It simplifies calculations significantly, providing a good approximation for long, thin charged objects where the length far exceeds the distance from the object at which the field is measured.

2. What if the line of charge is not infinitely long? The calculation becomes more complex and requires integration. Numerical methods are often used in such cases.

3. What are the units of linear charge density (λ)? The units are Coulombs per meter (C/m).

4. How does the direction of the electric field change if the line charge is negative? The direction reverses; it points towards the line of charge.

5. Are there limitations to this model? Yes, the "infinite" assumption breaks down near the ends of a finite line charge. The field near the ends will deviate from the 1/r dependence.

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1.6: Calculating Electric Fields of Charge Distributions 13 Jan 2021 · The electric field for a line charge is given by the general expression \[\vec{E}(P) = \dfrac{1}{4\pi \epsilon_0} \int_{\textrm{line}} \dfrac{\lambda dl}{r^2} \hat{r}. \nonumber\] A general element of the arc between \(\theta\) and \(\theta + d\theta\) is of length \(R\,d\theta\) and therefore contains a charge equal to \(\lambda R \,d\theta\).

Electric Field Due to a Line of Charge - Finite Length - Physics ... 6 Jan 2017 · It shows you how to evaluate the definite integrals using calculus techniques such as U-substitution and trigonometric substitution in order to derive the formula to calculate the net electric...

How to Determine the Electric Field of a Finite Line of Charge Learn how to determine the electric field of a finite line charge, and see examples that walk through sample problems step-by-step for you to improve your physics knowledge and skills.

Electric field due to a line charge - Physics Stack Exchange 21 Jul 2018 · Placing the origin of the cylindrical coordinate system $ (r,\phi,z)$ on the line of charge directly to the left of point $P$, then point $P$ is at $\vec {r}=r\,\hat {r}$. The line charge runs along the $z$-axis such that a general point on …

Electric field due to a line of charge - Physics Stack Exchange 8 Aug 2016 · In the case of an electric field, the direction is such that it always points away from the line of charge (in case the line of charge is positive). Why? Let us assume that the electric field wasn't exactly away from the line of charge and it bent towards the left.

Electric Field of a Line Charge - PHYSICS CALCULATIONS 21 Jul 2023 · To calculate the electric field intensity (electric field strength) produced by a line charge, we can employ Coulomb’s Law, which states that the electric field at a point is directly proportional to the charge and inversely proportional to the distance from the charge.

Electric Field of Line Charge - HyperPhysics Electric Field: Ring of Charge. The electric field of a ring of charge on the axis of the ring can be found by superposing the point charge fields of infinitesmal charge elements. The ring field can then be used as an element to calculate the electric field of a charged disc.

Electric Field Lines: Definition, Properties, and Drawings 2 Feb 2023 · English physicist Michael Faraday first developed the concept of electric field lines in the 1830s. For isolated charges, the electric field lines come out of a positive charge and terminate at infinity. On the other hand, they start from infinity and terminate at a negative charge.

23. THE ELECTRIC FIELD - University of Rochester Each field line starts on a positive point charge and ends on a negative point charge. Since the density of field lines is proportional to the strength of the electric field, the number of lines emerging from a positive charge must also be proportional to the charge. An example of field lines generated by a charge distributions is shown in ...

5.6: Electric Field Due to an Infinite Line Charge using Gauss’ Law 12 Sep 2022 · Use Gauss’ Law to determine the electric field intensity due to an infinite line of charge along the \(z\) axis, having charge density \(\rho_l\) (units of C/m), as shown in Figure \(\PageIndex{1}\).

5.6: Electric Field Due to an Infinite Line Charge using Gauss’ Law Use Gauss’ Law to determine the electric field intensity due to an infinite line of charge along the \(z\) axis, having charge density \(\rho_l\) (units of C/m), as shown in Figure \(\PageIndex{1}\).

Electric Field Due to Line Charge - Unacademy In this article, we will find the electric field due to a finite line charge at a perpendicular distance and discuss electric field line charge importance. We would find the electric field through the derivation method without using Gauss’s Law.

5.6: Calculating Electric Fields of Charge Distributions 15 Jan 2025 · The electric field for a line charge is given by the general expression \[\vec{E}(P) = \dfrac{1}{4\pi \epsilon_0} \int_{\textrm{line}} \dfrac{\lambda dl}{r^2} \hat{r}. \nonumber\] A general element of the arc between \(\theta\) and \(\theta + d\theta\) is of length \(R\,d\theta\) and therefore contains a charge equal to \(\lambda R \,d\theta\).

5.5 Calculating Electric Fields of Charge Distributions The electric field points away from the positively charged plane and toward the negatively charged plane. Since the σ σ are equal and opposite, this means that in the region outside of the two planes, the electric fields cancel each other out to zero. However, in the region between the planes, the electric fields add, and we get

Electric Field of Line Charge Calculator The electric field E of a line charge at a point located at a perpendicular distance r from the line of charge is given by the formula: E: This symbol represents the electric field at the point of interest. λ: This is the linear charge density, defined as the charge per unit length along the line of charge.

Electric Charges and Fields: Flux, Gauss Law & Coulomb's Law 19 Oct 2024 · Electric charge and field is one of the important topics of class 12th physics. ... The total number of electric field lines passing a given area in a unit of time is known as the electric flux. In electromagnetism, electric flux can be defined as the measure of the electric field via a given surface, even though an electric field cannot flow ...

3.3: Electrostatic Force - Coulomb's Law - Physics LibreTexts 27 Jan 2025 · The relevant distances and forces on charge 3 are shown below. Since the magnitude of the force on charge 1 by charge 4 is 4N, the same as the magnitude on charge 3 by charge 2 is also 4N, since the distance between 1 and 4 is the same as between 2 and 3, and the charges are identical, all positive with the same charge \(q\).

5.7: Electric Field Lines - Physics LibreTexts 15 Jan 2025 · Problem-Solving Strategy: Drawing Electric Field Lines. Electric field lines either originate on positive charges or come in from infinity, and either terminate on negative charges or extend out to infinity. The number of field lines originating or terminating at a charge is proportional to the magnitude of that charge.

Electric Field due to a Line Charge (contd.) Electric Field due to a ... Electric Field due to a Line Charge •Filament like distribution of charge density. •For example, sharp beam in a cathode-ray tube or charged conductor of a very small radius. •Let us assume an infinite straight-line charge, with charge density ρ l C/m, lying along the z …

Electric Field Due To Infinite Line Charge or Long Straight … Let us learn how to calculate the electric field due to infinite line charges. Consider an infinitely long straight, uniformly charged wire. Let the linear charge density of this wire be λ.

Electromagnetic field - Wikipedia An electromagnetic field (also EM field) is a physical field, mathematical functions of position and time, representing the influences on and due to electric charges. [1] The field at any point in space and time can be regarded as a combination of an electric field and a magnetic field.Because of the interrelationship between the fields, a disturbance in the electric field can create a ...

Electric Field Lines 7 Nov 2024 · Electric field lines around a point charge are directed away from a positive charge and towards a negative charge. A radial field spreads uniformly to or from the charge in all directions, but the strength of the field decreases with distance. The electric field is stronger where the lines are closer together

Electric field due to Line Charge - The Physicscatalyst Learn about concept and derivation of electric field due to finite line charge at equatorial point and electric field due to a line of charge at axial point.

Electric field from line charge equation | Example of Calculation 21 Mar 2024 · The electric field from a line charge is calculated using a derived equation based on Coulomb’s law and linear charge density. The line charge model is an idealized representation, but it is useful for approximating electric fields in many practical scenarios.