Problem 58 - University Physics with Modern Physics

Question

Infinite sheet \(A\) carries a positive uniform charge density \(\sigma\) , and sheet \(B\) , which is to the right of \(A\) and parallel to it, carries a uniform negative charge density \(-2 \sigma .\) (a) Sketch the electric field lines for this pair of sheets. Include the region between the sheets as well as the regions to the left of \(A\) and to the right of \(B\) . (b) Repeat part (a) for the case in which sheet \(B\) carries a charge density of \(+2 \sigma .\)

Step-by-Step Solution

Answer

Field lines between the sheets differ based on the charge of sheet B: attractively for \(-2\sigma\), canceling for \(+2\sigma\).

Step 1: Understanding Charge Densities

Sheet A has a positive uniform charge density \( \sigma \). Sheet B, to its right, has either \( -2 \sigma \) or \( +2 \sigma \) depending on the scenario. We need to determine the electric field lines for both configurations.

Step 2: Electric Field Lines for Sheet A

An infinite planar sheet with charge density \( \sigma \) generates a uniform electric field perpendicular to the sheet. The field points away from the sheet if the charge is positive, with a magnitude \( E = \frac{\sigma}{2\varepsilon_0} \). For sheet A, the field lines emerge perpendicularly from both sides of the sheet.

Step 3: Electric Field Lines for Sheet B with \(-2\sigma\)

For sheet B with \(-2\sigma\), the electric field lines will be twice as strong and point towards the sheet due to the negative charge. The magnitude is \( \frac{|-2\sigma|}{2\varepsilon_0} \). These lines will attract the field lines from sheet A in the region between the sheets.

Step 4: Sketching the Field Lines for Scenario with \(-2\sigma\)

Between sheets A and B, field lines will start at sheet A and end at sheet B, indicating the field direction from positive to negative sheets. To the left of A, field lines are directed outwards away from A, while to the right of B, lines enter from the right.

Step 5: Electric Field Lines for Sheet B with \(+2\sigma\)

If sheet B has \(+2\sigma\), its electric field is similar to sheet A but in the opposite direction. This results in double-strength field lines emerging outward from both sides of sheet B.

Step 6: Sketching the Field Lines for Scenario with \(+2\sigma\)

Between A and B, field lines from both sheets point in opposite directions, effectively canceling each other. To the left of A and right of B, the field lines diverge away from each sheet, showing regions of positive field addition.

Key Concepts

Charge Density

Charge density is a crucial concept in understanding electric fields, especially when dealing with infinite sheets. It is defined as the amount of charge per unit area, denoted by \(\sigma\). The positive or negative sign of charge density indicates whether the sheet has an excess of positive or negative charges, respectively.

• A positive charge density (\(\sigma \)) means the sheet has more positive charges.
• A negative charge density (\(-\sigma\)) implies more negative charges.

Charge density influences the strength and direction of electric fields. With infinite sheets, the distribution of charge is even, creating a uniform electric field.
Understanding charge density helps in predicting how electric field lines behave in various scenarios, such as when two sheets with differing charge densities are placed parallel to each other.

Infinite Sheets

When discussing electric fields, a common scenario involves infinite sheets. These are imaginary surfaces extending infinitely in two dimensions. Their endless nature simplifies calculations of electric fields, as edge effects are ignored.
The electric field generated by an infinite sheet is uniform and perpendicular to the surface, with the field's direction depending on the charge sign:

• For positive sheets, the field points away.
• For negative sheets, it points towards the sheet.

The magnitude of the field produced by an infinite sheet is given by the formula \(E = \frac{\sigma}{2\varepsilon_0}\). This formula shows that for an infinite sheet, the electric field magnitude depends solely on the charge density and not on the distance from the sheet.

Electric Field Lines

Electric field lines are a visual representation of how electric fields act in space. They provide insight into the field's direction and relative strength. Key properties of electric field lines include:

• Lines originate from positive charges (or infinite positive sheets).
• Lines terminate at negative charges (or infinite negative sheets).
• The closer the lines, the stronger the field.
• Lines never intersect one another.

In exercises involving infinite sheets, field lines allow us to gauge how sheets with different charge densities interact. For instance, in a system of two sheets, lines from a positive sheet extend outward, while those from a negative sheet reach inward, forming a pattern that can highlight regions of field interaction or cancellation.

Positive and Negative Charges

Positive and negative charges are fundamental to understanding electric fields. They determine how electric field lines are drawn between and around charged objects.

• Positive charges have electric field lines that radiate outward.
• Negative charges have lines that converge inward.

When two sheets, one with positive and one with negative charge density, are placed parallel, they create an attraction between field lines – lines extend from the positive sheet towards the negative sheet.
Conversely, when like charges face each other, like positive-positive, field lines push away, leading to cancellation in the area between the sheets, though the field outside remains significant.