Problem 60 - University Physics with Modern Physics

Question

Sketch the electric field lines for a disk of radius \(R\) with a positive uniform surface charge density \(\sigma\) . Use what you know about the electric field very close to the disk and very far from the disk to make your sketch.

Step-by-Step Solution

Answer

Near the disk, field lines are perpendicular and uniform; far from it, they spread like a point charge's field.

Step 1: Understand the Problem

We need to sketch the electric field lines for a disk with radius \(R\) and a positive uniform surface charge density \(\sigma\). To do this, let's use our understanding of electric fields both very close to and very far from such a charged disk.

Step 2: Analyze the Field Very Close to the Disk

When very close to the disk, the electric field lines will be perpendicular to the surface and uniformly distributed, similar to an infinite plane. This is due to the symmetry and uniform surface charge density of the disk. The electric field magnitude can be given approximately by \(E = \dfrac{\sigma}{2\varepsilon_0}\), where \(\varepsilon_0\) is the permittivity of free space.

Step 3: Analyze the Field Far from the Disk

When far from the disk, the behavior of the electric field changes. The disk can be approximated as a point charge, and the electric field lines will spread out radially from the disk as they would from a point charge. The field strength reduces following the inverse-square law, \(E = \dfrac{kQ}{r^2}\), where \(Q\) is the total charge on the disk and \(r\) is the distance from the disk.

Step 4: Combine Insights to Sketch the Field Lines

Start by drawing field lines perpendicular to the surface of the disk and evenly distributed across the disk when viewed from above. As you move away from the disk, the lines should begin to spread out spherically, similar to the field around a point charge. Ensure that lines are more closely spaced near the surface, indicating a stronger field, and spread out as they move further from the disk.

Key Concepts

Surface Charge Density

The concept of surface charge density is crucial in understanding electric field behavior around charged objects. Surface charge density, denoted by \( \sigma \), is defined as the amount of charge per unit area on a surface. It describes how charges are distributed across a surface, such as a disk, which helps us predict the electric field generated by the charged surface.

In our specific example with a disk, the uniform surface charge density means that charges are spread evenly. This kind of distribution leads to a uniform electric field very close to the surface of the disk, where the field lines are perpendicular and evenly spaced. A higher value of \( \sigma \) indicates more charge on the surface, which strengthens the electric field. Understanding this distribution is key to sketching electric field lines accurately, as it dictates how the field should appear both close to and farther away from the disk.

Permittivity of Free Space

Permittivity of free space, represented by the symbol \( \varepsilon_0 \), is a fundamental physical constant. It affects the behavior of electric fields in a vacuum and is essential in calculating electric fields around charged objects.

In the context of a charged disk, \( \varepsilon_0 \) helps determine the electric field strength close to the surface. When calculating the field strength near a charged plane or disk, we often use the formula \( E = \frac{\sigma}{2\varepsilon_0} \). This formula reflects the relationship between surface charge density and the resultant electric field. A larger \( \varepsilon_0 \) value would imply that the same surface charge density results in a weaker electric field, as the permittivity counteracts the effects of the charge. Understanding \( \varepsilon_0 \) is vital for predicting how field lines will behave in different environments and helps inform the correct sketching of electric field lines.

Inverse-Square Law

The inverse-square law is a fundamental concept describing how the strength of an electric field diminishes with distance. According to this law, the intensity of the field is inversely proportional to the square of the distance from the source of the field.

For a charged disk, when observed from a distance, we simplify the disk as a point charge. This results in the electric field line behavior following the inverse-square law, given by \( E = \frac{kQ}{r^2} \), where \( k \) is Coulomb's constant, \( Q \) is the total charge on the disk, and \( r \) is the distance from the disk.

• As you move away from the disk, the field strength decreases rapidly.
• This law explains why electric field lines spread out as we move farther away from a charged object.
• The lines are closer together near the charge, indicating stronger fields, and become more spaced apart with increasing distance.

This diminishing field strength nature is critical for drawing electric field lines as it shows how they transform from a concentrated radial pattern near the charge to a more dispersed distribution further away.

Electric Field Strength

Electric field strength is a measure of the force exerted by the field on a positive test charge placed within its vicinity. It's critical for determining how field lines should be drawn, reflecting their density and orientation.

For a disk with a uniform positive surface charge density, the electric field strength near the disk is given by \( E = \frac{\sigma}{2\varepsilon_0} \). This explains why, at close range, lines are tightly packed and perpendicular to the surface, symbolizing a strong and uniform field. As you distance yourself from the disk, you treat the entire disk as a point charge and field strength decreases following the inverse-square law formula, \( E = \frac{kQ}{r^2} \).

• Near the disk, field lines are concentrated, indicating higher field strength.
• As distance increases, lines diverge, showing decreasing field strength.
• These line patterns are crucial for visualizing how forces would act on charges within the field.

Understanding electric field strength helps create accurate sketches of electric field lines, ensuring they represent the true behavior of the field surrounding the charged disk.