A flat sheet of paper of area 0.250 m2 is oriented so that the normal to the sheet is at an angle of 60° to a uniform electric field of magnitude 14 N/C. (c) For what angle φ between the normal to the sheet and the electric field is the magnitude of the flux through the sheet (i) largest and (ii) smallest? Explain your answers.

Verified step by step guidance

1. Understand the concept of electric flux: Electric flux (Φ) through a surface is given by the formula Φ = E * A * cos(φ), where E is the electric field strength, A is the area of the surface, and φ is the angle between the normal to the surface and the direction of the electric field.
2. Identify the conditions for maximum flux: The cosine function, cos(φ), reaches its maximum value of 1 when φ = 0°. Therefore, the electric flux is largest when the angle between the normal to the sheet and the electric field is 0°, meaning the field is perpendicular to the sheet.
3. Identify the conditions for minimum flux: The cosine function reaches its minimum value of 0 when φ = 90°. Therefore, the electric flux is smallest when the angle between the normal to the sheet and the electric field is 90°, meaning the field is parallel to the sheet.
4. Explain why these angles result in maximum and minimum flux: When φ = 0°, the entire magnitude of the electric field contributes to the flux through the sheet, maximizing it. Conversely, when φ = 90°, the electric field is parallel to the sheet, and no field lines pass through it, minimizing the flux.
5. Summarize the findings: The angle φ for maximum flux is 0°, and for minimum flux is 90°. This is due to the behavior of the cosine function in the electric flux formula, which dictates how the orientation of the sheet relative to the electric field affects the flux.

Key Concepts

Electric Flux

Electric flux is a measure of the electric field passing through a given area. It is calculated as the product of the electric field magnitude, the area through which the field lines pass, and the cosine of the angle between the field lines and the normal to the surface. Mathematically, it is expressed as Φ = E * A * cos(θ), where E is the electric field, A is the area, and θ is the angle.

Angle Dependence of Flux

The angle between the normal to the surface and the electric field affects the electric flux. When the angle is 0°, the cosine function reaches its maximum value of 1, resulting in the maximum flux. Conversely, when the angle is 90°, the cosine function is 0, leading to zero flux. Thus, the flux is largest when the field is perpendicular to the surface and smallest when parallel.

Normal to a Surface

The normal to a surface is an imaginary line perpendicular to the surface at a given point. It is crucial in calculating electric flux because the angle used in the flux formula is measured between the electric field and this normal. Understanding the orientation of the normal helps in determining how the electric field interacts with the surface.