The formula for the electric field of an infinitely long, uniformly charged wire is E = λ / (2πrε₀).
Understanding the Formula
This formula describes the magnitude of the electric field (E) at a distance (r) from an infinitely long wire with a uniform linear charge density (λ). Let's break down each component:
- E: Represents the electric field strength. It's measured in Newtons per Coulomb (N/C). The electric field is a vector quantity, but this formula provides the magnitude of the field.
- λ: Represents the linear charge density, which is the amount of charge per unit length along the wire. It is measured in Coulombs per meter (C/m). A higher linear charge density will result in a stronger electric field.
- r: Represents the radial distance from the wire to the point where we are calculating the electric field. The electric field's magnitude decreases as you move farther away from the wire.
- ε₀: Represents the permittivity of free space (also known as the electric constant). It has a value of approximately 8.854 × 10⁻¹² F/m. This constant is a fundamental property of the vacuum and plays a critical role in determining the strength of electric fields.
Calculation of Electric Field
To calculate the electric field, you simply need to know the linear charge density (λ) and the radial distance (r) from the wire. Here is a simple breakdown of how to apply the formula, as described in the reference:
- Identify Variables: Note the linear charge density (λ) of the wire and the distance (r) at which you want to calculate the electric field. The permittivity of free space (ε₀) is a constant value.
- Apply the Formula: Substitute the given values for λ, r, and ε₀ in the equation E = λ / (2πrε₀).
- Calculate: Perform the calculation to find the electric field magnitude (E).
Practical Insights
- Infinite Wire Approximation: The formula assumes an infinitely long wire, which is, of course, a simplification. However, the formula is a very good approximation for the electric field around real-world wires when the distance (r) at which you calculate the electric field is significantly smaller than the length of the wire.
- Direction of the Field: The direction of the electric field is radial, pointing directly away from the wire if the charge density is positive and toward the wire if it is negative. In other words, if the wire has positive charge, the electric field lines will point away from the wire, and if the wire has negative charge, the electric field lines will point towards the wire.
- Cylindrical Symmetry: The formula also applies to the electric field of an infinitely long, uniformly charged cylinder, which exhibits cylindrical symmetry.
Example
Let's consider an example. Suppose a long wire has a linear charge density (λ) of 2.0 x 10⁻⁶ C/m. What is the electric field 0.5 meters away from the wire?
Using the formula:
E = λ / (2πrε₀)
E = (2.0 x 10⁻⁶ C/m) / (2π 0.5m 8.854 × 10⁻¹² F/m)
E ≈ 71,924 N/C
Therefore, the electric field at 0.5 meters from the wire is approximately 71,924 N/C.
