The earth has a vertical electric field at the surface, pointing down, that averages 100 N/C. This field is maintained by various atmospheric processes, including lightning. What is the excess charge on the surface of the earth?

Verified step by step guidance

1. Step 1: Understand the relationship between electric field and surface charge density. The electric field near a surface is related to the surface charge density (σ) by the equation: $E=\frac{\sigma }{\epsilon }$₀, where $E$ is the electric field, $\sigma$ is the surface charge density, and $\epsilon$₀ is the permittivity of free space (approximately $8.85\; \times \; 10-12C2/N·m2$). Rearrange this equation to solve for $\sigma$: $\sigma =E\epsilon$₀.
2. Step 2: Substitute the given value of the electric field into the equation. The electric field is given as $100\; N/C$. Use the permittivity of free space $\epsilon$₀ = 8.85 × 10^-12 C²/N·m². Calculate the surface charge density $\sigma$ using the formula: $\sigma =E\epsilon$₀.
3. Step 3: Relate the surface charge density to the total excess charge on the Earth's surface. The surface charge density $\sigma$ is defined as the charge per unit area: $\sigma =\frac{Q}{A}$, where $Q$ is the total charge and $A$ is the surface area. Rearrange this equation to solve for $Q$: $Q=\sigma A$.
4. Step 4: Use the approximate surface area of the Earth to calculate the total excess charge. The Earth's surface area is approximately $5.1\; \times \; 1014m2$. Substitute the calculated value of $\sigma$ from Step 2 and the Earth's surface area into the formula $Q=\sigma A$.
5. Step 5: Perform the multiplication to find the total excess charge $Q$. Ensure the units are consistent throughout the calculation (Coulombs for charge, square meters for area, etc.). The result will give the total excess charge on the Earth's surface.

Key Concepts

Electric Field

An electric field is a region around a charged object where other charged objects experience a force. It is defined as the force per unit charge and is measured in newtons per coulomb (N/C). In this context, the Earth's electric field of 100 N/C indicates the strength and direction of the force that a positive test charge would experience at the surface.

Gauss's Law

Gauss's Law relates the electric field at a surface to the charge enclosed by that surface. It states that the electric flux through a closed surface is proportional to the total charge enclosed within that surface. This principle is essential for calculating the excess charge on the Earth's surface by integrating the electric field over a hypothetical surface surrounding the Earth.

Surface Charge Density

Surface charge density is the amount of electric charge per unit area on a surface, typically expressed in coulombs per square meter (C/m²). It is crucial for determining the total excess charge on the Earth's surface when combined with the area over which the electric field acts. The relationship between electric field strength and surface charge density is given by the equation E = σ/ε₀, where ε₀ is the permittivity of free space.