Problem 57 - University Physics with Modern Physics

Question

Two horizontal, infinite, plane sheets of charge are separated by a distance \(d\) . The lower sheet has negative charge with uniform surface charge density \(-\sigma<0 .\) The upper sheet has positive charge with uniform surface charge density \(\sigma>0 .\) What is the electric field (magnitude, and direction if the field is nonzero) (a) above the upper sheet, (b) below the lower sheet, (c) between the sheets?

Step-by-Step Solution

Answer

(a) Zero, (b) Zero, (c) \(\frac{\sigma}{\varepsilon_0}\) downward.

Step 1: Understanding the Electric Field due to a Single Charged Sheet

The electric field due to an infinite plane sheet of charge with surface charge density \( \sigma \) is given by the formula \( E = \frac{\sigma}{2\varepsilon_0} \), where \( \varepsilon_0 \) is the permittivity of free space. If the sheet has a positive charge \( +\sigma \), the direction of the electric field is away from the sheet. If it has a negative charge \( -\sigma \), the direction of the electric field is towards the sheet.

Step 2: Calculating the Electric Field Above the Upper Sheet

Above the upper sheet, which is positively charged, the electric field directly due to the sheet is downward (away from the charge), with magnitude \( \frac{\sigma}{2\varepsilon_0} \). The lower negatively charged sheet is far enough apart and exerts an upward field of the same magnitude. These two fields cancel each other out.

Step 3: Calculating the Electric Field Below the Lower Sheet

Below the negatively charged lower sheet, the field due to the lower sheet is upward with magnitude \( \frac{\sigma}{2\varepsilon_0} \). The field due to the positively charged upper sheet at this position is directed downward, and again, it is of the same magnitude. These two fields cancel each other out.

Step 4: Calculating the Electric Field Between the Sheets

Between the sheets, the field from the positively charged upper sheet is directed downward with magnitude \( \frac{\sigma}{2\varepsilon_0} \), and the field from the negatively charged lower sheet is directed upward with the same magnitude. Hence, the fields add up (both pointing downward), giving a total electric field of magnitude \( \frac{\sigma}{\varepsilon_0} \) directed downward.

Key Concepts

Electric Field

Understanding the concept of an electric field is crucial in electrostatics. The electric field is a vector quantity represented as \ \( E \ \), which describes the force per unit charge exerted on a positive test charge placed in the vicinity of other charges. In the context of an infinite plane sheet with uniform surface charge density \ \( \sigma \ \), the electric field's magnitude is given by the formula \\[ E = \frac{\sigma}{2\varepsilon_0} \\]\where \ \( \varepsilon_0 \ \) is the permittivity of free space.
The direction of the electric field is determined by the type of charge on the sheet:

• For a positive charge \ \( \sigma>0 \ \), the field is directed away from the sheet.
• For a negative charge \ \( -\sigma<0 \ \), the field is directed towards the sheet.

Understanding how these fields interact when sheets are arranged, like in the exercise, is crucial to calculate the net electric field in regions surrounding and between the sheets.

Surface Charge Density

Surface charge density, symbolized as \ \( \sigma \ \), refers to the amount of electric charge per unit area on a surface. It is measured in units of coulombs per square meter \ \( \text{C/m}^2 \ \). This concept is essential when analyzing electric fields generated by charged surfaces.
An infinite plane sheet of charge creates a uniform electric field perpendicular to the surface. The value and sign of \ \( \sigma \ \) significantly influence the direction and strength of this field:

• A positive surface charge density \ \( \sigma>0 \ \) means charges are spread across the surface creating an outward pull.
• A negative surface charge density \ \( \sigma<0 \ \) makes the field point inward towards the sheet.

In electrostatic problems, understanding these interactions helps predict how combined fields impact surrounding areas, revealing cancellation or enhancement effects depending on respective charge configurations.

Permittivity of Free Space

The permittivity of free space, represented by the symbol \ \( \varepsilon_0 \ \), is a fundamental constant of nature that characterizes the ability of a vacuum to permit electric field lines. Its approximate value is \ \( 8.85 \times 10^{-12} \, \text{F/m} \ \) (farads per meter). This constant plays a crucial role in electrostatics and influences the strength of forces between charges.
In formulas describing electric fields, \ \( \varepsilon_0 \ \) appears as a denominator, signifying its function in moderating electric interactions. Specifically, for an infinite plane sheet:

• The electric field magnitude is inversely proportional to \ \( \varepsilon_0 \ \), highlighting that a greater permittivity weakens the influence of a given charge density.

This constant is vital for accurately computing electric phenomena in theoretical and practical setups, ensuring that the predictions align with physical reality.