Long uniform line of charge. A very long straight wire
possesses a uniform positive charge per unit length λ, Calculate the electric field at points near (but outside) the wire, far from the ends.
To calculate the electric field at points near (but outside) a long uniform line of charge, such as a very long straight wire with a uniform positive charge per unit length (λ), you can use the concept of Gauss's law.
Let's assume that the wire is aligned along the z-axis, and the point of interest is at a distance 'r' from the wire, where r is much larger than the length of the wire.
Here's how you can calculate the electric field at that point:
1. Choose a Gaussian surface: Consider a cylindrical Gaussian surface with radius 'r' and length 'L'. The Gaussian surface should enclose the wire but not be too close to the ends.
2. Calculate the total charge enclosed: Since the wire has a uniform charge per unit length (λ), the total charge enclosed by the Gaussian surface is given by Q = λL.
3. Apply Gauss's law: Gauss's law states that the electric flux through a closed surface is proportional to the total charge enclosed. Mathematically, it is given by Φ = EA = Q/ε₀, where Φ is the electric flux, E is the electric field, A is the area of the Gaussian surface, Q is the total charge enclosed, and ε₀ is the vacuum permittivity.
4. Calculate the electric field: The electric field is perpendicular to the Gaussian surface and has the same magnitude at all points along the surface. Hence, you can consider a circular cross-section of the Gaussian surface at a distance 'r' from the wire.
The electric flux passing through this circular cross-section is Φ = EA = E(2πrL).
Since the electric field lines are perpendicular to the cylindrical surface, the electric field does not contribute to the flux through the curved surface of the Gaussian cylinder - only the top and bottom surfaces contribute.
Thus, the total electric flux is given by Φ = 2EπrL.
Setting this equal to Q/ε₀, we get:
2EπrL = λL/ε₀
Simplifying this equation, we find:
E = λ/(2πε₀r)
Therefore, the electric field at a distance 'r' from the wire (outside the wire, far from the ends) is given by E = λ/(2πε₀r).
You can use this formula to calculate the electric field based on the given value of λ and the distance 'r' from the wire.