Question

Calculate the maximum surface charge distribution that can be maintained on any surface surrounded by dry air.

Short answer

Answer: The maximum surface charge density that can be maintained on any surface surrounded by dry air is approximately 5.31 x 10^-5 C/m².

Step-by-step solution

Find the maximum electric field in dry air before dielectric breakdown

The maximum electric field in dry air, also known as the dielectric strength, is approximately 3 x 10^6 N/C. This value represents the electric field when the air's insulating properties break down, and an electric discharge occurs.

Identify the relationship between surface charge density and electric field

The electric field near the surface of a charged flat conductor is given by E = σ / (2ε₀), where E is the electric field, σ is the surface charge density, and ε₀ is the vacuum permittivity (8.854 x 10^-12 C^2/Nm^2).

Solve for surface charge density, σ

Rearranging the formula from Step 2, we have σ = 2Eε₀. Substituting the values for E (from Step 1) and ε₀, we get: σ = 2 (3 x 10^6 N/C) (8.854 x 10^-12 C²/Nm²) = 5.31 x 10^-5C/m²

Interpret the result

The maximum surface charge density that can be maintained on any surface surrounded by dry air is approximately 5.31 x 10^-5 C/m². This is the maximum charge density that can be sustained before the electric field in air becomes strong enough to cause dielectric breakdown, leading to an electric discharge.

Key concepts

Dielectric Breakdown

Dielectric breakdown is a phenomenon that occurs when an insulating material, like dry air, becomes electrically conductive. This change happens when the electric field within the material reaches a critical level, known as the dielectric strength. At this point, the insulating properties of the material are compromised, and an electrical discharge can occur, which might result in sparks or full-scale arcing.

For instance, when looking at dry air, its dielectric strength is about 3 x 10^6 newtons per coulomb (N/C). When designing circuits or electrical components, engineers must ensure that the electric field never exceeds this threshold to prevent damage and maintain safety. In the case of our exercise, understanding dielectric breakdown is essential to calculate the maximum surface charge distribution on a surface surrounded by dry air without causing an electrical discharge.

Electric Field

An electric field is a region around a charged particle or object within which a force would be exerted on other charged particles or objects. The strength of the electric field is measured in volts per meter (V/m) or newtons per coulomb (N/C).

The concept of an electric field is crucial when it comes to determining the surface charge density that can be held on an object. For conductors, the electric field is perpendicular to the surface and its magnitude is directly related to the surface charge density. The larger the charge on the surface, the stronger the electric field produced, and vice versa. As noted in the exercise, this relationship helps us calculate the maximum surface charge density before reaching the threshold for dielectric breakdown in surrounding air.

Surface Charge Density

Surface charge density (\(\text{symbolized as } \sigma \)) is the amount of charge per unit area on a surface. For a conductor, we can relate the electric field (\(E\)) near its surface to the surface charge density using the formula \(E = \frac{\sigma}{2 \varepsilon_0}\), where \(\varepsilon_0\) represents the vacuum permittivity. This formula enables us to determine the amount of electrical charge that can be distributed over a conductor's surface before reaching a level at which the surrounding insulator (air, in our textbook problem) would fail.

In our example, solving for the surface charge density gives us the maximum amount that can be held without leading to dielectric breakdown. Using the relationship between the surface charge density and the electric field, one can derive practical insights for the design and safety of electronic components.

Vacuum Permittivity

Vacuum permittivity (\(\varepsilon_0\)), also known as the electric constant, is a fundamental physical constant which characterizes the ability of the vacuum to permit electric field lines. Its value is approximately 8.854 x 10^-12 coulombs squared per newton meter squared (C²/Nm²).

While the term 'vacuum permittivity' might suggest a relevance only to the vacuum, this constant is also crucial for calculations involving other mediums, such as air. For example, in the exercise scenario, vacuum permittivity is a vital component of the formula that ties together electric field strength and surface charge density on a conductor. It essentially dictates how much electric field is generated by a certain amount of surface charge in air. Hence, this constant plays a key role in understanding the electrical behavior of materials and is extensively used in electrostatics calculations.