Professional Courses
Industry-relevant training in Business, Technology, and Design to help professionals and graduates upskill for real-world careers.
Categories
Interactive Games
Fun, engaging games to boost memory, math fluency, typing speed, and English skillsβperfect for learners of all ages.
Typing
Memory
Math
English Adventures
Knowledge
Enroll to start learning
Youβve not yet enrolled in this course. Please enroll for free to listen to audio lessons, classroom podcasts and take practice test.
Interactive Audio Lesson
Listen to a student-teacher conversation explaining the topic in a relatable way.
Introduction to Electric Field from a Charged Wire
Unlock Audio Lesson
Signup and Enroll to the course for listening the Audio Lesson
Today, we're going to dive into the electric field created by an infinitely long, straight uniformly charged wire. Can anyone tell me what happens to the electric field around a charged object?
I think the electric field radiates outward from the object.
Exactly! In the case of a charged wire, the field is not just in any direction; it radiates uniformly around the wire. This symmetry plays a crucial role in our analysis.
But why do we care about infinitely long wires? Isnβt that unrealistic?
Good question! While no wire is truly infinite, considering an infinitely long wire helps us understand the electric field far away from the ends of the wire, where edge effects become negligible.
Now, can anyone think of a memory aid for remembering how the electric field behaves around charged objects?
Maybe we could think of it like rays of sunlight! They spread out in all directions but stay equally spaced as they move away from the source.
That's a perfect analogy! Just like sunlight, the electric field lines represent the strength and direction of the field at various points around the wire.
Using Gauss's Law
Unlock Audio Lesson
Signup and Enroll to the course for listening the Audio Lesson
Letβs apply Gauss's law to find the electric field around our charged wire. Who remembers what Gauss's law states?
It says that the total electric flux through a closed surface is proportional to the charge enclosed.
Exactly! By taking a cylindrical Gaussian surface around our wire, we can simplify our calculations. Why do we choose this shape?
Because it's symmetric! Each part of the cylinder is at the same distance from the wire.
Right again! This symmetry allows us to conclude that the electric field is constant over the curved surface. Can anyone calculate the area of this cylindrical side?
The area is 2Οrh, where h is the height of the cylinder.
Great! So now we can express the flux in terms of the electric field and this area. Let's summarize: We find that **E Γ 2Οrh = charge enclosed / Ξ΅β**.
Understanding the Result
Unlock Audio Lesson
Signup and Enroll to the course for listening the Audio Lesson
Now that we have our flux expression, let's derive the electric field. Can someone explain the charge enclosed if our linear charge density is Ξ»?
The total charge would be Ξ» multiplied by the length of the wire we are considering, right?
Absolutely! That gives us the total charge in terms of Ξ» and the height of the cylinder. Plugging this back in gives us our electric field formula!
**E = Ξ» / (2ΟΞ΅βr)** Wow, thatβs a clean result!
Exactly! This shows the electric field is dependent on the distance r from the wire. It also helps us visualize how the strength of the field diminishes as we move away from the wire.
So does this mean that a closer wire exerts a stronger field?
Yes! As you get closer, the influence of the wire's charge becomes stronger, reflecting in the electric fieldβs magnitude.
Introduction & Overview
Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.
Quick Overview
Standard
It explores the properties and implications of the electric field produced by such a wire, including the symmetry of the field and the use of Gauss's law to derive the field's magnitude.
Detailed
Field due to an Infinitely Long Straight Uniformly Charged Wire
This section examines the electric field created by an infinitely long straight wire that carries a linear charge density, denoted as Ξ». By utilizing Gauss's law, the section calculates the electric field, demonstrating that it is radial and does not vary with the position along the wireβs length.
The setup considers points around the wire, illustrating how they possess mutual symmetry in relation to the charged wire. Consequently, the direction of the electric field is always radial β outward if the wire is positively charged and inward if it is negatively charged.
To determine the field strength, a cylindrical Gaussian surface is employed. Since the electric field is uniform across the curved surface of the cylinder, the flux through the closed surface is given as the product of the electric field and the curved surface area of the cylinder. The charge enclosed by the Gaussian surface is equal to the linear charge density multiplied by the length of the wire that the surface encloses. Then, by applying Gauss's law, the electric field can be expressed mathematically as:
E = Ξ» / (2ΟΞ΅βr)
where E represents the electric field, Ξ΅β the permittivity of free space, and r the radial distance from the wire. This formula highlights that the electric field strength decreases inversely with distance from the wire.
Youtube Videos
Audio Book
Dive deep into the subject with an immersive audiobook experience.
Symmetry and Direction of Electric Field
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
Consider an infinitely long thin straight wire with uniform linear charge density l. The wire is obviously an axis of symmetry. Suppose we take the radial vector from O to P and rotate it around the wire. The points P, PΒ’, PΒ’Β’ so obtained are completely equivalent with respect to the charged wire. This implies that the electric field must have the same magnitude at these points. The direction of electric field at every point must be radial (outward if l > 0, inward if l < 0).
Detailed Explanation
In this portion, we are discussing a long straight wire carrying a uniform charge. Since the wire is straight and infinite, it makes sense to consider it symmetric. If we imagine looking at this wire from any angle, the electric field would look the same at any radial distance from the wire, which means that the electric field strength only depends on how far you are from the wire and not on where you are along the length of the wire. The direction of the electric field will point away from the wire if it's positively charged and towards it if negatively charged.
Examples & Analogies
Think of the electric field as the air moving around a straight stick that is infinitely long in both directions. No matter where you are around the stick, if you look straight out away from it, the amount of air you feel pushing against you (the electric field strength) is the same at every point that is the same distance from the stick.
Calculating the Electric Field
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
To calculate the field, imagine a cylindrical Gaussian surface, as shown in the Fig. 1.26(b). Since the field is everywhere radial, flux through the two ends of the cylindrical Gaussian surface is zero. At the cylindrical part of the surface, E is normal to the surface at every point, and its magnitude is constant, since it depends only on r. The surface area of the curved part is 2prl, where l is the length of the cylinder. Flux through the Gaussian surface = E Γ 2prl. The surface includes charge equal to ll. Gaussβs law then gives E Γ 2prl = ll/e0.
Detailed Explanation
In this portion, we are using Gauss's law to calculate the electric field produced by the charged wire. We first visualize an imaginary cylindrical surface that surrounds the wire (this is our Gaussian surface). Because the electric field is radial, we know that there's no field passing through the ends of our cylinderβonly through its curved surface. When we calculate the flux (the amount of electric field passing through an area) through the curved surface, we can express it as E times the area of the curved surface (which is 2prl). Using Gauss's law, we relate this flux to the total charge enclosed by the Gaussian surface, which simplifies down to the electric field equation.
Examples & Analogies
Imagine wrapping a piece of string around a hot dog. If we think of the hot dog as the charged wire and the string as the Gaussian surface, we can see how the heat (the electric field) radiates outward from the hot dog. If you were to calculate how much heat you feel per area as you hold the hot dog, you would find it increased the closer you got to it, much like the electric field increases as you get closer to the charged wire.
Final Expression for Electric Field
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
From Gaussβs law, we find that E = Ξ»/(2ΟΞ΅0r) where Ξ» is the linear charge density, Ξ΅0 is the permittivity of free space, and r is the distance from the wire. The electric field vector is given by E = (Ξ»/(2ΟΞ΅0r)) nΜ where nΜ is the outward radial unit vector.
Detailed Explanation
Finally, we derive the electric field produced by the charged wire. By substituting the expressions for the flux and the charge enclosed back into Gauss's law, we arrive at a clear formula for E that shows how it changes with respect to the distance from the wire and how it relates to the charge per unit length on the wire. This gives us both the magnitude and direction of the electric field.
Examples & Analogies
Picture standing in a large cylindrical room where the walls are covered in party balloons. The balloons are the charges, and they give off a gentle push (the electric field) that is strongest closest to the wall and gets weaker as you walk away from it. If we measured how strongly this push felt at different distances from the wall, we could create a graph just like the one we derived here, illustrating how the 'push' (electric field) decreases as you increase your distance (r).
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
-
Gauss's Law: Describes the relationship between electric flux and charge enclosed.
-
Electric Field: Describes the force per unit charge exerted by a charged object.
-
Symmetry: Key property enabling the simplification of electric field calculations.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
-
For an infinitely long charged wire, the electric field at a distance r is given by E = Ξ» / (2ΟΞ΅βr). This equation shows how the strength of the field reduces inversely with distance from the wire.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
π΅ Rhymes Time
-
Charged wire is what we see, spreading fields so radially.
π Fascinating Stories
-
Imagine a wizard casting light from a line of magic, each point glowing brighter as you draw closer.
π§ Other Memory Gems
-
Remember: 'E-L-P', for Electric field, Linear charge density, depends on Proximity.
π― Super Acronyms
'E-L of WIRE'
- Electric field from Long wire Is Radial & Extends.
Flash Cards
Review key concepts with flashcards.
Glossary of Terms
Review the Definitions for terms.
-
Term: Electric field
Definition:
A field around charged particles that exerts forces on other charged particles.
-
Term: Gaussian surface
Definition:
An imaginary closed surface used in Gauss's law to calculate electric flux.
-
Term: Linear charge density (Ξ»)
Definition:
Charge per unit length along a charged line.
-
Term: Radial field
Definition:
An electric field that radiates outward from a charge.
-
Term: Symmetry
Definition:
A property that allows simplifications in the calculation of forces in similar configurations.