Problem 52 - University Physics with Modern Physics

Question

A very long, straight wire has charge per unit length \(1.50 \times 10^{-10} \mathrm{C} / \mathrm{m} .\) At what distance from the wire is the electric- field magnitude equal to 2.50 \(\mathrm{N} / \mathrm{C}\)?

Step-by-Step Solution

Answer

The distance is approximately 1.08 cm.

Step 1: Understand the Problem

We are given a long, straight wire with a linear charge density \(\lambda = 1.50 \times 10^{-10} \, \mathrm{C/m}\) and we need to find the distance \(r\) from the wire where the electric field \(E\) is equal to \(2.50 \, \mathrm{N/C}\).

Step 2: Recognize the Electric Field Formula for a Line Charge

The electric field due to a long straight wire at a distance \(r\) is given by: \[ E = \frac{\lambda}{2 \pi \varepsilon_0 r} \] where \(\varepsilon_0\) is the permittivity of free space and its value is \(8.85 \times 10^{-12} \, \mathrm{C^2/(N\cdot m^2)}\).

Step 3: Plug in the Values and Rearrange for r

Substitute \(E = 2.50 \, \mathrm{N/C}\) and \(\lambda = 1.50 \times 10^{-10} \, \mathrm{C/m}\) into the formula: \[ 2.50 = \frac{1.50 \times 10^{-10}}{2 \pi \times 8.85 \times 10^{-12} \times r} \] Rearranging for \(r\), we have: \[ r = \frac{1.50 \times 10^{-10}}{2 \pi \times 8.85 \times 10^{-12} \times 2.50} \]

Step 4: Calculate r

Perform the calculation: \[ r = \frac{1.50 \times 10^{-10}}{2 \pi \times 8.85 \times 10^{-12} \times 2.50} \] Using a calculator: \[ r \approx 1.08 \times 10^{-2} \, \mathrm{m} \]

Step 5: Conclusion

The distance from the wire where the electric field magnitude is \(2.50 \, \mathrm{N/C}\) is approximately \(1.08 \times 10^{-2} \, \mathrm{m}\), or \(1.08 \, \mathrm{cm}\).

Key Concepts

Linear Charge Density

Linear charge density, represented as \( \lambda \), is a measure of the amount of electric charge per unit length along a line, such as a wire.
It is an important concept when dealing with problems related to electric fields generated by line charges.

• It's defined in terms of coulombs per meter (C/m).
• In the given problem, the wire has a linear charge density \( \lambda = 1.50 \times 10^{-10} \mathrm{C/m} \).

Understanding linear charge density helps determine how charge is distributed along a wire, which directly affects the electric field's magnitude and direction around the wire.

Long Straight Wire

A long straight wire is a common model used in physics to simplify the calculation of electric fields.
This model assumes the wire is infinitely long, which allows certain symmetries to simplify mathematical calculations.

• By modeling a very long wire, the electric field can be treated as uniform in a plane perpendicular to the wire.
• This assumption allows us to ignore edge effects, making calculations easier.

In the context of the given problem, the simplification of an infinite wire helps in applying Gauss's law easily to find the electric field.

Permittivity of Free Space

The permittivity of free space, \( \varepsilon_0 \), is a fundamental constant that characterizes how electric fields interact with the vacuum.
Its value is \( 8.85 \times 10^{-12} \mathrm{C^2/(N\cdot m^2)} \), making it crucial for calculating electric fields in a vacuum or air.

• It acts as a proportionality factor in the formulas for electric fields, linking electric field intensity to charge density.
• In the electric field equation \( E = \frac{\lambda}{2 \pi \varepsilon_0 r} \), \( \varepsilon_0 \) directly influences the field strength.

Understanding \( \varepsilon_0 \) is key to solving problems related to electric fields in spaces where matter is negligible.

Electric Field Formula

The electric field formula for a long straight wire provides a way to compute the electric field created by the wire at a certain distance.
This formula is derived using Gauss's law, which relates the electric field to the charge enclosed by a Gaussian surface.

• For a linear charge, the formula is \( E = \frac{\lambda}{2 \pi \varepsilon_0 r} \).
• The variables \( \lambda \) and \( r \) denote linear charge density and the distance from the wire, respectively.
• \( \varepsilon_0 \) is the permittivity of free space and acts as a scaling factor.

By applying this formula, we can determine the strength of the electric field at various points around the wire, showing how distance and charge density affect the field.