A hemispherical surface with radius r in a region of uniform electric field E→ has its axis aligned parallel to the direction of the field. Calculate the flux through the surface.

Verified step by step guidance

1. Identify the given parameters: the radius of the hemispherical surface (r) and the uniform electric field (E).
2. Recall the formula for electric flux, which is given by Φ = E * A * cos(θ), where A is the area through which the field lines pass and θ is the angle between the field lines and the normal to the surface.
3. Note that the electric field is parallel to the axis of the hemisphere, so θ = 0 degrees, and cos(0) = 1.
4. Calculate the area of the flat circular surface of the hemisphere, which is A = π * r^2.
5. Substitute the values into the flux formula: Φ = E * π * r^2.

Key Concepts

Electric Flux

Electric flux is a measure of the quantity of electric field lines passing through a given surface. It is mathematically defined as the dot product of the electric field vector and the area vector of the surface, integrated over the surface. The formula for electric flux (Φ) is given by Φ = ∫ E · dA, where E is the electric field and dA is the differential area element. Understanding electric flux is crucial for solving problems related to electric fields and Gauss's law.

Gauss's Law

Gauss's Law relates the electric flux through a closed surface to the charge enclosed by that surface. It states that the total electric flux through a closed surface is equal to the enclosed charge divided by the permittivity of free space (ε₀). This law is fundamental in electrostatics and simplifies the calculation of electric fields for symmetric charge distributions. It provides a powerful tool for analyzing electric fields in various geometries.

Uniform Electric Field

A uniform electric field is one in which the electric field strength is constant in both magnitude and direction throughout a region. This means that the field lines are parallel and evenly spaced. In the context of the problem, the hemispherical surface is placed in such a field, which simplifies the calculation of electric flux since the angle between the electric field and the surface normal can be easily determined. Understanding the characteristics of uniform electric fields is essential for analyzing their effects on charged objects.