A spark occurs at the tip of a metal needle if the electric field strength exceeds 3.0×1063.0\$$times10^6 N/C$$, the field strength at which air breaks down. What is the minimum surface charge density for producing a spark?

Understand the relationship between electric field strength and surface charge density. The electric field near the surface of a conductor is related to the surface charge density (σ) by the formula: E=σ/ε₀, where ε₀ is the permittivity of free space. Identify the given values in the problem. The electric field strength at which air breaks down is 3.0 × $$10^{6} N/C$$, and the permittivity of free space is a constant: ε₀=8.85×10⁻12 C²/N·m². Rearrange the formula to solve for surface charge density (σ). Using E=σ/ε₀, multiply both sides by ε₀ to isolate σ: σ=E·ε₀. Substitute the known values into the formula. Replace E with 3.0 × $$10^{6} N/C $$and ε₀ with 8.85 × $$10^{-12} C$$²/N·m². Perform the multiplication to find the minimum surface charge density (σ). The result will be in units of C/m², which represents the surface charge density required to produce a spark.

Ch 24: Gauss' Law

Knight Calc - Physics for Scientists and Engineers 5th Edition

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Knight Calc 5th Edition

Ch 24: Gauss' Law

Problem 26

Chapter 24, Problem 26

A spark occurs at the tip of a metal needle if the electric field strength exceeds $3.0 \times 10^{6}$ N/C, the field strength at which air breaks down. What is the minimum surface charge density for producing a spark?

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Understand the relationship between electric field strength and surface charge density. The electric field near the surface of a conductor is related to the surface charge density (σ) by the formula:

E = σ / ε ₀

, where

ε ₀

is the permittivity of free space.

Identify the given values in the problem. The electric field strength at which air breaks down is

3.0 × 10

⁶ N/C, and the permittivity of free space is a constant:

ε ₀ = 8.85 \times 10 ⁻ 12 C ² / N \cdot m ²

Rearrange the formula to solve for surface charge density (σ). Using

E = σ / ε ₀

, multiply both sides by

ε ₀

to isolate σ:

σ = E \cdot ε ₀

Substitute the known values into the formula. Replace

E

with

3.0 × 10

⁶ N/C and

ε ₀

with

8.85 × 10

^-12 C²/N·m².

Perform the multiplication to find the minimum surface charge density (σ). The result will be in units of

C/m²

, which represents the surface charge density required to produce a spark.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Electric Field Strength

Electric field strength is a measure of the force per unit charge experienced by a charged particle in an electric field. It is expressed in newtons per coulomb (N/C) and indicates how strong the electric field is at a given point. In this context, the electric field strength of 3.0 x 10⁶ N/C is the threshold at which air becomes ionized, allowing a spark to occur.

Surface Charge Density

Surface charge density refers to the amount of electric charge per unit area on a surface, typically measured in coulombs per square meter (C/m²). It is crucial for determining how much charge is needed on the needle's surface to create an electric field strong enough to exceed the breakdown threshold of air, leading to a spark.

Breakdown Voltage

Breakdown voltage is the minimum voltage that causes a portion of an insulator to become electrically conductive. In the case of air, when the electric field strength exceeds 3.0 x 10⁶ N/C, the air molecules become ionized, allowing current to flow and resulting in a spark. Understanding this concept is essential for calculating the necessary surface charge density to achieve this electric field strength.

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A spark occurs at the tip of a metal needle if the electric field strength exceeds 3.0×1063.0\(\times\)10^6 N/C, the field strength at which air breaks down. What is the minimum surface charge density for producing a spark?

Verified video answer for a similar problem:

Key Concepts

Electric Field Strength

Surface Charge Density

Breakdown Voltage

A spark occurs at the tip of a metal needle if the electric field strength exceeds $3.0 \times 10^{6}$ N/C, the field strength at which air breaks down. What is the minimum surface charge density for producing a spark?