Question

A negative point charge \(-Q\) is situated near a large metal plate that has a total charge of \(+Q .\) Sketch the electric field lines.

Short answer

Field lines emerge perpendicularly from the metal plate and curve towards the negative point charge, converging onto it.

Step-by-step solution

Understanding the Problem

We have a point charge \(-Q\) and a large metal plate with a total charge of \(+Q\). Since the metal plate is large, it can be considered as an infinite charged sheet. Our task is to sketch the electric field lines between and around these two objects.

Analyzing the Charges

The negative point charge \(-Q\) will have electric field lines converging towards it, as electric field lines point from positive to negative charges. The positively charged metal plate, acting like an infinite sheet, will have electric field lines emerging perpendicularly from its surface.

Sketching Electric Field Lines for the Plate

The electric field lines emerging from the metal plate will be perpendicular to its surface, since it acts like an infinite charged sheet. These lines are evenly spaced and uniformly distributed over the surface of the plate, indicating an electric field directed away from the plate.

Sketching Electric Field Lines for the Point Charge

The electric field lines around the negative point charge \(-Q\) converge towards the charge. These lines are radially inward pointing from all directions towards the charge. As we get closer to the charge, the field lines become denser.

Combining the Electric Field Lines

In the region between the point charge \(-Q\) and the metal plate, the electric field lines from the plate will curve and converge towards the negative point charge, forming a pattern where they are redirected to the charge. Around the charge and away from the plate, field lines will still respect the radial convergence towards the \(-Q\) point charge.

Key concepts

Point Charge

A point charge is a charge that is concentrated at a single point in space. In reality, no charge is truly a point, but by considering a charge as a point, we can simplify the analysis. This concept is useful when dealing with electric fields, as the effect of the charge's concentrated location can be easily studied. The key characteristics of a point charge are:

• It has a definite charge value, like \(-Q\), representing either a positive or negative charge.
• The electric field due to a point charge radiates radially outward if positive, and radially inward if negative.
• The strength of the electric field generated by a point charge is described by Coulomb's law, such that the field decreases with the square of the distance from the charge.

These properties make point charges an essential part of electrostatics and help visualize how charges interact with each other.

Metal Plate

A metal plate in electrostatics is typically considered a conductor. Conductors have free electrons that can move, which is crucial for understanding how these materials interact with electric fields. When a metal plate is charged, it can be studied similarly to an infinite charged sheet:

• Charges on a metal plate will distribute evenly across the surface due to repulsion between like charges.
• The electric field lines emerge perpendicularly from the surface of the charged plate.
• If the plate is part of a problem, like in our situation with the plate having charge \(+Q\), we assume it's large enough that edge effects (perturbations at the edges of the plate) are negligible.

Understanding the behavior of charged metal plates helps us explain how charges interact with other nearby objects, such as point charges.

Electric Field Lines

Electric field lines are a visual tool used to represent the strength and direction of an electric field surrounding charges. These lines have several important properties:

• Lines begin at positive charges and end at negative charges.
• The density of these lines represents the strength of the electric field; closer lines mean a stronger field.
• Field lines never cross each other since that would imply two directions of the field at one point, which is impossible.
• The field lines are perpendicular to the surface of conductors.

In practical terms, sketching electric field lines helps us understand the interaction between different charges and charged surfaces. They help answer questions like where fields are strongest and how charges will move in an electric field.

Charged Sheet

Imagine a charged sheet as an infinite expanse of charge distributed evenly across a flat surface. It's a theoretical concept, perfect for simplifying problems. Understanding a charged sheet involves a few key principles:

• The electric field created by an infinite charged sheet is uniform and independent of the distance from the sheet, given by \(E = \frac{\sigma}{2\varepsilon_0}\), where \(\sigma\) is surface charge density.
• Field lines extend perpendicularly from the surface of the sheet, reflecting its infinite nature and uniform charge distribution.
• This uniform field model is used to approximate the behavior of large charged surfaces like metal plates, where edge effects can be ignored.

Charged sheets are a foundational concept in electrostatics and help to explain how uniform fields arise from large, flat surfaces.