Problem 25 - Fundamentals of Physics

Question

SSM An infinite line of charge produces a field of magnitude \(4.5 \times\) \(10^{4} \mathrm{~N} / \mathrm{C}\) at distance \(2.0 \mathrm{~m}\). Find the linear charge density.

Step-by-Step Solution

Answer

The linear charge density is \( 5.0 \times 10^{-7} \mathrm{~C/m} \).

Step 1: Understand the Relation Between Electric Field and Linear Charge Density

For an infinite line of charge, the electric field, \( E \), at a distance \( r \) from the line is given by the formula \( E = \frac{\lambda}{2 \pi \varepsilon_0 r} \), where \( \lambda \) is the linear charge density, and \( \varepsilon_0 \) is the permittivity of free space, approximately \( 8.85 \times 10^{-12} \mathrm{~C}^2/\mathrm{N} \cdot \mathrm{m}^2 \). Our goal is to solve for \( \lambda \).

Step 2: Re-arrange the Formula to Solve for Linear Charge Density

Rearrange the formula to solve for \( \lambda \):\[\lambda = 2 \pi \varepsilon_0 r E\]

Step 3: Substitute Known Values into the Formula

Substitute the given values into the equation: \( r = 2.0 \mathrm{~m} \), \( E = 4.5 \times 10^{4} \mathrm{~N/C} \), \( \varepsilon_0 = 8.85 \times 10^{-12} \mathrm{~C}^2/\mathrm{N} \cdot \mathrm{m}^2 \):\[\lambda = 2 \pi (8.85 \times 10^{-12}) (2.0) (4.5 \times 10^{4})\]

Step 4: Calculate the Linear Charge Density

Perform the calculations:\[\lambda = 2 \pi \cdot 8.85 \times 10^{-12} \cdot 2.0 \cdot 4.5 \times 10^{4} = 5.0 \times 10^{-7} \mathrm{~C/m}\]

Step 5: Verify Units and Dimensions

Ensure that the units in the calculations match up: - The units for \( \varepsilon_0 \) are \( \mathrm{C}^2/\mathrm{N} \cdot \mathrm{m}^2 \), which correctly cancel with the electric field \( \mathrm{N/C} \) to give charge per unit length in \( \mathrm{C/m} \).

Key Concepts

Infinite Line of Charge

An infinite line of charge is a theoretical concept often used in physics to simplify calculations. Imagine it like a long wire stretching infinitely in both directions, uniformly charged along its entire length. In reality, no physical wire is infinitely long, but for distances sufficiently far from the ends, a finite wire can be approximated as infinite.

• The important property of an infinite line of charge is its electric field.
• The field is uniform, meaning it doesn't change with distance along the length of the line.
• It only depends on the radial distance from the line.
• This is different from points charges, where the field magnitude decreases with the square of the distance.

The simplified geometry means that the electric field at any point perpendicular to the line is only radially directed. The mathematical relation simplifies to a cylindrical symmetry. Studying an infinite line of charge is fundamental in electromagnetism, helping to understand how charges distribute and how electric fields behave.

Linear Charge Density

Linear charge density, represented by the symbol \( \lambda \), is a way of describing the distribution of electric charge along a line. It measures how much charge exists per unit length of the line. This concept is particularly useful for understanding properties of charged lines, like wires or fibers.

• Linear charge density is expressed in units of coulombs per meter (\( \text{C/m} \)).
• It helps in calculating the electric field created by long, charged objects.
• In large systems, it simplifies calculations compared to considering individual charges.

When you know the linear charge density of an object, you can easily calculate the total charge on a segment of the line by multiplying \( \lambda \) by the segment's length. For finite and infinitely long charged objects, it is a key parameter to determine how they influence their surroundings through their electric field.

Permittivity of Free Space

The permittivity of free space, also known as the electric constant \( \varepsilon_0 \), is a fundamental physical constant that quantifies the ability of a vacuum to permit electric field lines. It plays a crucial role in electromagnetism, defining how electric fields interact with materials and space.

• Its approximate value is \( 8.85 \times 10^{-12} \text{ C}^2/\text{N} \cdot \text{m}^2 \).
• It appears in Coulomb's law and the equation for capacitance.
• In SI units, it is part of the equation used to calculate the electric force between two charges.

The permittivity of free space essentially dictates how strongly electric fields interact in a vacuum situation. In problems involving charges in vacuum or air, \( \varepsilon_0 \) is pivotal in determining field strengths and forces between charges. Understanding \( \varepsilon_0 \) allows scientists and engineers to design and analyze electrical systems effectively.