A flat sheet of paper of area 0.2500.2500.250 m2 is oriented so that the normal to the sheet is at an angle of 606060° to a uniform electric field of magnitude 141414 N/C. For what angle ϕ\phi 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.

Ch 22: Gauss' Law

Young & Freedman Calc - University Physics 15th Edition

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Young & Freedman Calc 15th Edition

Ch 22: Gauss' Law

Problem 1c

Chapter 22, Problem 1c

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. 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

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.

Identify the conditions for maximum and minimum flux: The cosine function, cos(φ), determines the variation of flux with angle φ. The flux is largest when cos(φ) is maximum, which occurs at φ = 0° (cos(0°) = 1). The flux is smallest when cos(φ) is minimum, which occurs at φ = 90° (cos(90°) = 0).

Explain why the flux is largest at φ = 0°: When the angle φ is 0°, the normal to the sheet is aligned with the electric field direction, maximizing the component of the field passing through the sheet. This results in the largest possible flux.

Explain why the flux is smallest at φ = 90°: At φ = 90°, the normal to the sheet is perpendicular to the electric field direction, meaning no component of the field passes through the sheet. This results in zero flux.

Summarize the findings: The magnitude of the electric flux through the sheet is largest when the angle φ is 0° and smallest when the angle φ is 90°, due to the properties of the cosine function in the electric flux formula.

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Key Concepts

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

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 it passes, and the cosine of the angle between the field and the normal to the surface. This concept helps determine how much of the field interacts with the surface.

Angle Dependence of Flux

The angle between the electric field and the normal to the surface affects the electric flux. When the angle is 0°, the flux is maximized because the field is perpendicular to the surface, allowing maximum interaction. Conversely, at 90°, the flux is minimized as the field is parallel to the surface, resulting in no interaction.

Cosine Function in Flux Calculation

The cosine function is crucial in calculating electric flux, as it accounts for the angle's effect on the field's interaction with the surface. The flux is proportional to the cosine of the angle, meaning it decreases as the angle increases from 0° to 90°. Understanding this relationship is key to predicting flux behavior at different angles.

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Textbook Question

You measure an electric field of $1.25 \times 10^{6}$ N/C at a distance of $0.150$ m from a point charge. There is no other source of electric field in the region other than this point charge.

(a) What is the electric flux through the surface of a sphere that has this charge at its center and that has radius $0.150$ m?

(b) What is the magnitude of this charge?

Textbook Question

A charged paint is spread in a very thin uniform layer over the surface of a plastic sphere of diameter $12.0$ cm, giving it a charge of $−49.0$ $μ$ C. Find the electric field just outside the paint layer;

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Textbook Question

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.

(a) Find the magnitude of the electric flux through the sheet.

(b) Does the answer to part (a) depend on the shape of the sheet? Why or why not?

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Textbook Question

A charged paint is spread in a very thin uniform layer over the surface of a plastic sphere of diameter $12.0$ cm, giving it a charge of $negative 49.0$ $μ$ C. Find the electric field just inside the paint layer.

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