Problem 51 - College Physics

Question

\(\bullet\) \(\bullet\) Sketch electric field lines in the vicinity of two charges, \(Q\) and \(-4 Q,\) located a small distance apart on the \(x\) -axis.

Step-by-Step Solution

Answer

Draw field lines from \(Q\) towards \(-4Q\), ensuring more lines terminate at \(-4Q\) due to its greater magnitude.

Step 1: Identifying charges and their location on the x-axis

First, recognize that we have two point charges: a positive charge denoted as \(Q\) and a negative charge \(-4Q\). These charges are located a small distance apart on the x-axis. The exact locations are not specified, but for clarity, suppose \(Q\) is at \(x = -a\) and \(-4Q\) is at \(x = a\).

Step 2: Understanding electric field line behavior around charges

Remember that electric field lines originate from positive charges and terminate at negative charges. The strength of the field is indicated by the density of lines. Since \(-4Q\) is stronger, more lines should terminate at it compared to those originating from \(Q\).

Step 3: Drawing field lines: starting at Q

Start drawing the electric field lines from the positive charge \(Q\). These lines should extend outward radially. Since there is a significant negative charge \(-4Q\), some lines should curve towards \(-4Q\).

Step 4: Drawing field lines: ending at -4Q

From the negative charge \(-4Q\), draw lines that attract the lines coming from \(Q\) and ensure there are more lines coming into \(-4Q\) than from \(Q\), showing it has a larger magnitude. These lines should also extend towards infinity, outside the area of influence of \(Q\).

Step 5: Refining the diagram with field line properties

Verify that the diagram respects the properties of electric fields: field lines never cross, and their density reflects the strength of the field. The central area between \(Q\) and \(-4Q\) should have lines bending inward towards \(-4Q\) to show attraction.

Key Concepts

Point Charges

Point charges are essential in understanding electric fields because they serve as the origin or conclusion of electric field lines. A point charge is simply a model of a particle with an electrical charge, idealized as a point in space. These charges can be positive or negative.

• Positive Point Charge ( Q ): A source from which electric field lines originate. These are shown by lines radiating outward.
• Negative Point Charge ( -4Q ): A sink where electric field lines converge. The lines move toward this charge, demonstrating attraction.

Understanding how these point charges interact is key to sketching the field lines. In this scenario, a positive charge ( Q ) and a negative charge ( -4Q ) are near each other on the x-axis. They impact how electric fields are visualized, with field lines emerging from the positive charge and terminating at the more intense negative charge.

Electric Field Behavior

Electric field behavior helps to predict how electric forces act in space around charges. Electric field lines are a useful tool to visualize this behavior. They show how a positive test charge would move within the field.

• Originating from Positive Charge: Electric field lines begin at a positive charge, radiating outwards uniformly if isolated.
• Terminating at Negative Charge: These lines head towards a negative charge, demonstrating the attractive force.
• Interaction Shape: When charges are near, lines from the positive charge curve towards the negative, showing the direction of force.

The presence of a stronger charge, such as -4Q relative to Q , means more field lines will terminate at -4Q than originate from Q . This forms a visual map of electric forces at play between the two charges.

Electric Field Line Density

The density of electric field lines provides a visual indication of the field's strength. A denser region of lines implies a stronger field, whereas sparse lines indicate a weaker field.

• Field Strength Proportionality: The number of lines drawn corresponds to the relative magnitude of the charge. More lines signify stronger fields.
• Charge Magnitude Influence: Since -4Q is stronger than Q , a higher density of lines will illustrate its stronger pull.
• Regional Density: In areas between the charges, lines may appear denser, showcasing the interaction between them.

In practical diagrams, field line density helps quickly convey the difference in charge magnitude and the areas of stronger electric interaction. This core concept helps in both understanding the nature of electric forces and predicting the outcomes of charge interactions in space.