Electric field from line charge equation

Explore the electric field equation for line charge distributions, its derivation, and an example calculation. Learn key concepts in electrostatics.

Understanding the Electric Field from Line Charge Equation

When studying electrostatics, one of the key concepts is the electric field produced by various charge distributions. In this article, we will specifically discuss the electric field produced by a line charge, and the equation that governs it.

The Line Charge Model

A line charge is a distribution of electric charge along a straight, infinitely long line. This is an idealized model, as in reality, there are no true infinitely long lines of charge. However, it serves as a useful approximation in many practical scenarios, especially when the length of the charged line is much larger than the distance from the line to the point of interest.

Deriving the Electric Field Equation

To derive the equation for the electric field produced by a line charge, we begin with Coulomb’s law. Coulomb’s law states that the electric force between two point charges is proportional to the product of the charges and inversely proportional to the square of the distance between them:

1. F_E = kQq/r²

Where F_E is the electric force, k is the electrostatic constant, Q and q are the magnitudes of the charges, and r is the distance between the charges. However, to calculate the electric field E, we need to consider a unit charge q:

1. E = F_E/q = kQ/r²

Now, we need to express the charge distribution along the line using linear charge density (λ), which represents the charge per unit length. To do this, we must integrate the electric field produced by infinitesimal charge elements (dq) over the entire length of the line charge:

1. E = k ∫ (λ dx) / r²

Upon evaluating this integral and considering the geometry of the problem, we arrive at the final equation for the electric field produced by a line charge:

1. E = 2kλ/r

Key Takeaways

• The electric field from a line charge is calculated using a derived equation based on Coulomb’s law and linear charge density.
• The line charge model is an idealized representation, but it is useful for approximating electric fields in many practical scenarios.
• The final equation for the electric field produced by a line charge is E = 2kλ/r, where E is the electric field, k is the electrostatic constant, λ is the linear charge density, and r is the distance from the line charge.

In conclusion, understanding the electric field produced by a line charge is crucial when studying electrostatics. The equation E = 2kλ/r provides a solid foundation for analyzing the behavior of electric fields around line charge distributions, and is an essential tool for solving problems in this area of physics.

Example Calculation of Electric Field from a Line Charge

Let’s consider a practical example to demonstrate the calculation of the electric field produced by a line charge. Suppose we have a straight line charge with a linear charge density λ = 3 x 10^-9 C/m. We want to calculate the electric field strength E at a distance r = 0.1 m perpendicular to the line charge.

To do this, we will use the equation derived in the previous section:

1. E = 2kλ/r

First, we need to know the value of the electrostatic constant k. The electrostatic constant, also known as Coulomb’s constant, is given by:

1. k = 8.9875 x 10⁹ N·m²/C²

Now, we can plug in the values for k, λ, and r into the equation to calculate the electric field E:

1. E = 2(8.9875 x 10⁹ N·m²/C²)(3 x 10^-9 C/m) / (0.1 m)

After simplifying and performing the calculation, we find the electric field strength E:

1. E ≈ 5.39 x 10⁵ N/C

In this example, the electric field strength produced by the line charge with a linear charge density of 3 x 10^-9 C/m at a distance of 0.1 m perpendicular to the line charge is approximately 5.39 x 10⁵ N/C.