Problem 44 - University Physics with Modern Physics

Question

A very large plastic sheet carries a uniform charge density of \(-6.00 \mathrm{nC} / \mathrm{m}^{2}\) on one face. (a) As you move away from the sheet along a line perpendicular to it, does the potential increase or decrease? How do you know, without doing any calculations? Does your answer depend on where you choose the reference point for potential? (b) Find the spacing between equipotential surfaces that differ from each other by 1.00 V. What type of surfaces are these?

Step-by-Step Solution

Answer

(a) The potential increases as you move away from the sheet because the electric field points towards the sheet. (b) Find the electric field magnitude; the spacing for a 1 V difference is proportional to \( \frac{1.00}{E} \).

Step 1: Electric Field and Electric Potential Direction

The electric field produced by a uniformly charged infinite plane sheet is constant in magnitude and points away from the sheet if the charge is positive, or towards the sheet if the charge is negative. Since the charge density is negative, the electric field is directed towards the sheet. Therefore, as you move away from the sheet, you go against the direction of the electric field, which means the electric potential increases.

Step 2: Reference Point for Electric Potential

The change in electric potential is determined by the work done against the electric field. While the absolute value of potential depends on your reference point, the sign of change in potential, whether it increases or decreases, does not depend on the reference point.

Step 3: Calculate the Electric Field Magnitude

For an infinite plane with charge density \( \sigma = -6.00 \text{ nC/m}^2 \), the magnitude of the electric field is given by \( E = \frac{|\sigma|}{2\varepsilon_0} \), where \( \varepsilon_0 \) is the permittivity of free space \( 8.85 \times 10^{-12} \text{ C}^2/\text{N} \cdot \text{m}^2 \). Compute the magnitude of \( E \) using \( \sigma = 6.00 \times 10^{-9} \text{ C/m}^2 \).

Step 4: Calculate the Spacing Between Equipotential Surfaces

The potential difference \( \Delta V \) between two points is given by \( \Delta V = Ed \) for a uniform electric field, where \( d \) is the distance between equipotential surfaces. To find the spacing where \( \Delta V = 1.00 \text{ V} \), rearrange to get \( d = \frac{1.00}{E} \). Substitute the value of \( E \) from Step 3 to find \( d \).

Step 5: Identify the Type of Surfaces

Equipotential surfaces in a uniform electric field are parallel to each other and perpendicular to the electric field lines. For a plane sheet, the equipotential surfaces are planes parallel to the charged sheet.

Key Concepts

Electric Field

Electric fields are essential concepts in understanding how charges interact. For a large, uniformly charged infinite plane sheet, the electric field is uniform and constant in magnitude at any point away from the sheet. The direction of the electric field depends on the charge of the sheet:

• Positive charge on the sheet means the electric field points away from the sheet.
• Negative charge, as in our original exercise, causes the electric field to point towards the sheet.

This directionality is key to determining how the electric potential changes as you move further or closer along a line perpendicular to the sheet.
Since the electric field direction is towards the sheet due to the negative charge, moving away from the sheet is against the field. This means you are doing work against the electric field, which results in an increase in electric potential.

Equipotential Surfaces

Equipotential surfaces are fascinating features that simplify our understanding of electric fields. These surfaces are imaginary planes where every point holds the same electric potential. In the context of the charged infinite plane:

• Equipotential surfaces are parallel to the sheet.
• They are always perpendicular to the direction of the electric field.

This perpendicular nature is crucial because if you move along an equipotential surface, there is no change in electric potential, meaning no work is required. In a uniform field, like near an infinite charged sheet, equipotential surfaces maintain a constant separation, in alignment with the field's uniformity.
This constancy aids in visualizing that moving from one surface to the next involves a specific potential difference, calculated using the relationship between electric field strength and distance.

Charge Density

Charge density is a measure of how much electric charge is accumulated over a specific area. For a plane sheet, the charge density is represented by the symbol \( \sigma \) in units like Coulombs per square meter (C/m^2). Negative values, such as in our problem, indicate negative charges spread across the face of the sheet.
With a unit charge distribution like \( \sigma = -6.00 \text{ nC/m}^2 \), this results in an electric field being created around the sheet that influences both the direction and the magnitude of electric potential changes as you move away from it. Charge density directly impacts:

• The strength of the electric field due to the relationship \( E = \frac{|\sigma|}{2\varepsilon_0} \).
• The behavior of equipotential surfaces, as a denser charge will produce more closely spaced surfaces.

Understanding charge density is vital for predicting how electric fields and potentials interact in given configurations.

Infinite Plane Sheet

An infinite plane sheet is a theoretical construct that helps simplify complex electrostatic problems by assuming the sheet extends infinitely in all directions. This assumption leads to:

• A uniform and constant electric field everywhere perpendicular to the sheet.
• Simplified potential calculations as field lines remain parallel and normal.

When applied to the charged plane, such as our reference sheet with \( \sigma = -6.00 \text{ nC/m}^2 \), the infinite nature simplifies solving for potential differences and field strengths. The sheet's theoretical infinity ensures that edge effects, common in real sheets, are negligible, ensuring accuracy over large distances.
Understanding this concept helps when evaluating or designing systems where uniform fields are advantageous, such as in capacitors or shielding applications.