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Physics

Electric Field Due to an Infinitely Long Straight Uniformly Charged Wire

Electric Field Due to an Infinitely Long Straight Uniformly Charged Wire

Q: 6. What are the units of the electric field in this context?

The electric field (E) is measured in Newtons per Coulomb (N/C) or equivalently, Volts per meter (V/m).

How to Derive the Electric Field of an Infinite Straight Uniformly Charged Wire Using Gauss’s Law?

The topic of Electric Field Due To An Infinitely Long Straight Uniformly Charged Wire is important in physics and helps us understand how electric fields behave around line charges. This concept is essential for Boards, JEE, and NEET exams, and it forms the foundation for understanding more complex systems and real-world cables or conductor arrangements.

Understanding Electric Field Due To An Infinitely Long Straight Uniformly Charged Wire

Electric Field Due To An Infinitely Long Straight Uniformly Charged Wire refers to the electric field produced at any point in space due to a straight wire that carries a constant linear charge density (λ), and is assumed to be infinitely long for symmetry. It plays a vital role in topics like electric flux, Gauss Theorem, and electrostatics involving continuous charge distributions.

Formula or Working Principle of Electric Field Due To An Infinitely Long Straight Uniformly Charged Wire

The electric field due to an infinitely long straight uniformly charged wire is derived using Gauss's Law. The symmetry of the infinite wire allows us to use a cylindrical Gaussian surface, making the derivation straightforward. The standard formula is:

E = λ / (2πε₀r)

where E is the electric field at a distance r from the wire, λ is the linear charge density, and ε₀ is the permittivity of free space. The field points radially outward (for positive charge) or inward (for negative charge) and decreases as 1/r.

Here’s a useful table to understand Electric Field Due To An Infinitely Long Straight Uniformly Charged Wire better:

Electric Field Due To An Infinitely Long Straight Uniformly Charged Wire Table

Concept	Description	Example
Linear Charge Density (λ)	Charge per unit length of the wire	λ = 2 × 10^-6 C/m
Gaussian Surface	Cylindrical surface used to apply Gauss's Law	Surrounding the wire with radius r
Electric Field (E)	Field at distance r from wire, E = λ/(2πε₀r)	At r = 0.1 m, E is calculated using the formula

Worked Example / Practical Experiment

Let’s solve a problem step by step:

1. Identify the known values:
λ = 2 × 10^-6 C/m, r = 0.1 m, ε₀ = 8.854 × 10^-12 C²/N·m²

2. Apply the formula:
E = λ / (2π ε₀ r)

3. Substitute values:
E = (2 × 10^-6) / (2 × 3.1416 × 8.854 × 10^-12 × 0.1)
E ≈ (2 × 10^-6) / (5.561 × 10^-12)
E ≈ 3.6 × 10⁵ N/C

4. Analyze:
The result shows a strong electric field close to the wire, decreasing as r increases.

Conclusion: This approach helps apply electric field due to an infinitely long straight uniformly charged wire in real scenarios, like understanding cable insulation requirements.

Practice Questions

Define electric field due to an infinitely long straight uniformly charged wire with an example.
Write the formula for electric field due to an infinitely long straight uniformly charged wire.
How does the field depend on the distance from the wire and why?
Explain the working principle behind using a cylindrical Gaussian surface.

Common Mistakes to Avoid

Misinterpreting the formula—mistaking 1/r for 1/r² dependency (which applies for point charges).
Incorrectly choosing the Gaussian surface shape (must be cylindrical).
Forgetting to include 2π in the denominator.
Confusing linear charge density (λ) with surface or volume charge density.

Real-World Applications

Electric field due to an infinitely long straight uniformly charged wire is widely used in fields like cable design, transmission lines, and particle accelerators. It's also crucial for understanding shielded cables, electrostatic forces in devices, and modelling approximate behaviors of long conductors. Vedantu helps you connect such physics concepts with real-world applications and exam questions.

In this article, we explored Electric Field Due To An Infinitely Long Straight Uniformly Charged Wire—its meaning, derivation, practical application, and importance in physics. Keep exploring related physics topics with Vedantu to strengthen your fundamentals.

Related topics you may find useful: Electric Flux, Gauss Theorem, Unit of Electric Field, Electric Charge, Electric Potential Point Charge, and Electrostatics.

FAQs on Electric Field Due to an Infinitely Long Straight Uniformly Charged Wire

1. What is the electric field due to an infinitely long straight uniformly charged wire?

The electric field due to an infinitely long, uniformly charged straight wire is a radial field whose magnitude is directly proportional to the linear charge density (λ) of the wire and inversely proportional to the distance (r) from the wire. This is derived using Gauss's Law. The formula is E = λ / (2πε₀r), where ε₀ is the permittivity of free space.

2. How do you derive the electric field of an infinitely long wire using Gauss's Law?

Gauss's Law states that the total electric flux through a closed surface is proportional to the enclosed charge. For an infinitely long wire, we choose a cylindrical Gaussian surface. The electric field is parallel to the surface's sides, and the flux through the ends is zero due to symmetry. The flux through the curved surface is then used, along with the enclosed charge, to derive the formula E = λ / (2πε₀r).

3. What is linear charge density (λ) in the context of an infinitely long wire?

Linear charge density (λ) represents the amount of charge per unit length along the wire. It's typically expressed in Coulombs per meter (C/m). A higher λ indicates a greater charge density and thus a stronger electric field.

4. How is the Gaussian surface chosen for this derivation, and why is this choice crucial?

A cylindrical Gaussian surface is chosen because of the symmetry of the electric field around the infinitely long wire. The electric field lines are radial and perpendicular to the curved surface of the cylinder. This symmetry simplifies the calculation of the electric flux. Choosing a different surface would make the calculation significantly more complex.

5. What is the dependence of the electric field on the distance from the wire?

The electric field is inversely proportional to the distance (r) from the wire. This means that the field strength decreases as you move further away from the wire. Specifically, it follows a 1/r relationship, unlike the 1/r² relationship for a point charge.

6. What are the units of the electric field in this context?

The electric field (E) is measured in Newtons per Coulomb (N/C) or equivalently, Volts per meter (V/m).

7. Why doesn't the electric field of an infinite wire follow the inverse square law?

Unlike a point charge, which creates a field spreading out in all three dimensions, an infinitely long wire's field spreads only in two dimensions (perpendicular to the wire's length). This geometrical difference leads to the inverse proportionality with distance (1/r) instead of the inverse square (1/r²) law.

8. How is the electric field direction determined near the infinitely long wire?

The electric field lines are radial and point directly away from a positively charged wire. If the wire is negatively charged, the field lines point towards the wire. The direction is always perpendicular to the wire at any point.

9. What are common mistakes students make when applying this formula?

Common mistakes include: incorrect use of units (C/m, N/C, ε₀), forgetting the factor of 2 in the denominator, misinterpreting the direction of the electric field, and incorrectly applying the formula to situations where the wire is not infinitely long or the charge is not uniformly distributed.

10. Can I use this formula for real-world, finite-length wires?

The formula for an infinitely long wire provides a good approximation for the electric field near the center of a long, straight wire, provided that the distance from the wire is significantly smaller than the wire's length. For points close to the ends, the approximation is less accurate.

11. What if the charge distribution on the wire isn't uniform?

If the charge distribution is not uniform, the formula E = λ / (2πε₀r) is no longer valid. The calculation becomes much more complex and typically requires integration techniques to determine the electric field at a given point.

12. How does this concept relate to other electrostatic principles?

This concept is fundamentally linked to Gauss's Law, the principle of superposition for electric fields, and the concept of electric potential. Understanding these interconnected principles provides a comprehensive grasp of electrostatics.