Potential of Infinite Wire

In summary, the conversation discusses the potential difference between a point at a distance of b from the center of an infinitely long wire and the surface of the wire, as well as the electric field inside the conductor. Gauss's Law is applied to find the electric field, with the correct solution being E = ρ_Sa/ε_0b.
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Homework Statement


[itex]\rho_{wire}=a[/itex], surface charge density [itex]\rho_S[/itex]

What is potential difference of a point a distance of [itex]b[/itex] measured from the centre of the infinitely long wire, and the surface of the wire?

Since all of the charge is dispersed on the surface of the conductor, there will exist no charge within the surface, and henceforth the electric field inside the conductor is zero. In addition, symmetry of the problem tells us that the electric field radiates outwards from the conductor.

Let the origin be centred in the wire, such that the z-axis runs parallel to the infinitely long wire. We can apply Gauss's Law utilizing a gaussian surface such that [itex]\rho_{gaussian}=g > a[/itex], and the height from the xy-plane will be [itex]l[/itex].

At this point, I just want to know if I am solving for the electric field properly. Thanks in advance.

We know that
[tex]\frac{1}{ε_o}\int _V \rho dV=\oint_S\vec{E}\cdot\vec{dS}[/tex]
[tex]\frac{\rho_S(2\pi al)}{ε_o}=\oint_S\vec{E}\cdot\vec{dS}=E\int_SdS=E(2\pi gl)\rightarrow E = \frac{\rho_Sa}{b}[/tex],
 
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  • #2
The only mistake you made was to omit ε0 in the denominator.
 

1. What is an infinite wire and what are its properties?

An infinite wire is a hypothetical model used in physics to study the behavior of an infinitely long, straight wire with no endpoints. It has several properties, including zero resistance, uniform current distribution, and a magnetic field that surrounds the wire in a circular pattern.

2. How is the potential of an infinite wire calculated?

The potential of an infinite wire can be calculated using the formula V = μ0 I / 2πr, where V is the potential, μ0 is the permeability of free space, I is the current in the wire, and r is the distance from the wire.

3. What is the significance of the potential of an infinite wire in electromagnetism?

The potential of an infinite wire is important in understanding the concept of electric potential and how it relates to the electric field and charges in a system. It also helps in analyzing the behavior of electric fields around wires and in different configurations.

4. Can an infinite wire exist in reality?

No, an infinite wire is a theoretical model used for simplification and is not physically possible. Real wires have finite lengths and endpoints, and are subject to resistance and other physical limitations.

5. How does the potential of an infinite wire change with distance?

The potential of an infinite wire decreases as the distance from the wire increases. This is because the electric field strength decreases with distance, and the potential is directly proportional to the electric field. As the distance approaches infinity, the potential approaches zero.

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