The formula for the magnitude of the electric field due to an infinite line charge is E = λ / (2πϵ₀r).
This formula, derived from Gauss's Law, is a cornerstone of electrostatics and is crucial for understanding how electric fields are generated by charged objects. The following sections break down the formula and provide further context.
Understanding the Components of the Formula
The formula E = λ / (2πϵ₀r) can be further explained by examining each component:
- E: Represents the magnitude of the electric field at a point a distance r away from the line charge. The electric field is a vector quantity but this equation provides its magnitude.
- λ (lambda): This represents the linear charge density of the infinite line charge. Linear charge density indicates how much charge is distributed per unit length of the line. It is measured in Coulombs per meter (C/m).
- ϵ₀ (epsilon naught): This is the permittivity of free space, also known as the electric constant. It is a fundamental physical constant with an approximate value of 8.854 × 10⁻¹² F/m (Farads per meter). Permittivity describes a medium's ability to permit electric field lines.
- r: This variable represents the perpendicular distance from the line charge to the point at which the electric field is being measured.
Practical Implications and Insights
- Field Direction: The direction of the electric field due to a line charge is always radial, either pointing directly away from a positively charged line, or directly towards a negatively charged line.
- Distance Dependence: The electric field's magnitude diminishes as the distance r from the line charge increases. Specifically, it decreases inversely proportional to r.
- Uniform Charge Distribution: This formula assumes a uniformly charged infinite line. In reality, infinite line charges do not exist; however, this model can be a good approximation for very long, charged objects at points that are significantly close compared to the length of the line.
- Applications: This formula is useful in scenarios involving objects that can be approximated as long, charged lines, such as thin wires.
Example Calculation
Let's consider an example to illustrate how to use this formula.
Example:
Suppose we have an infinite line charge with a linear charge density (λ) of 5 x 10-6 C/m. We want to determine the magnitude of the electric field at a distance of 2 meters away from the line charge.
-
Identify the Knowns:
- λ = 5 x 10-6 C/m
- r = 2 m
- ϵ₀ = 8.854 × 10⁻¹² F/m
-
Apply the Formula:
E = λ / (2πϵ₀r) = (5 × 10⁻⁶ C/m) / (2π × 8.854 × 10⁻¹² F/m × 2 m) -
Calculate the Result:
E ≈ 44,940 N/C
Therefore, the magnitude of the electric field at a distance of 2 meters from the infinite line charge is approximately 44,940 Newtons per Coulomb.
Summary
The electric field due to a uniformly charged infinite line is defined by the formula E = λ / (2πϵ₀r), where λ is the linear charge density and r is the radial distance to the point of observation. This formula is important in numerous applications in physics and electrical engineering.
