Electric potential due to long wire

In summary, the problem is to find an expression for the electric potential at a distance r from an infinitely long straight line of charge of length h with linear charge density lambda per unit length, using Gauss' law or another method. The solution involves using the equation E = -grad*V and evaluating it at an arbitrary point a. However, the boundary condition that V = 0 at infinity must be assumed, as the charge is infinite due to the infinite length of the wire.
  • #1
vs5813
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Homework Statement


Using Gauss' law, or otherwise, find an expression for electric potential at a distance r from an infinitely long straight line of charge of length h with linear charge density lambda per unit length.

Homework Equations



q=lambda*h


The Attempt at a Solution


The first part is easy, using gauss' law to find electric field..the result i get is

E=lambda/(2*pi*r*epsilon_0)

..which should be correct. But then I'm not sure how to find potential in this case. I tried using the equation:

E= - grad*V

so

V = integral[E*dl]

V= -lambda/(2*pi*epsilon_0) integral_from_infinity_to_r[(1/r) dr]

..which leads me to:

V = -lambda/(2*pi*epsilon_0) [ln(r)]

where the ln(r) is evaluated from infinity to r...and I am not sure that looks right, or where to go from there.. :confused: I don't know what the solution is supposed to be and I couldn't find any explanations when i googled it..anyone have any idea? :uhh: thankyou sooo much!
 
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  • #2
Hello.

In fact that's the solution. That's life!
 
  • #3
heh :redface: thankyou!
 
  • #4
how do you evaluate ln at infinity and r??
 
  • #5
pick an arbitrary point a, and integrate with respect to that point. you're correct that you can't integrate to an infinite potential at the wire.
 
  • #6
the only way i can think of is that u have to assume the boundary condition that V=0 at infinity. but then that seems forced because when evaluating the similar case for a sphere u get a 1/r term and when u plug in infinity there it goes to zero (see griffiths 3rd ed example 2.6). can anyone reconcile these cases? i don't see any explanation above addressing this.
 
  • #7
Hi,

You cannot use infinity, because the problem states that there is an infinite length of wire. This tells us that the charge is infinite. Using a variable like "a" is your best bet for solving the problem.
 

What is "electric potential due to long wire"?

"Electric potential due to long wire" refers to the electric potential created by a long, straight wire carrying an electric current. This potential is created by the charges in the wire and is measured in volts.

How is the electric potential due to a long wire calculated?

The electric potential due to a long wire can be calculated using the formula V = kλ/r, where V is the potential, k is the Coulomb's constant, λ is the linear charge density of the wire, and r is the distance from the wire. This formula assumes that the wire is infinitely long and the charges are evenly distributed.

What factors affect the electric potential due to a long wire?

The electric potential due to a long wire is affected by the length of the wire, the amount of current flowing through it, and the distance from the wire. Additionally, the type of material the wire is made of and any external electric fields can also impact the potential.

How does the electric potential due to a long wire relate to electric fields?

The electric potential due to a long wire is directly related to electric fields. As an electric field is created by the charges in the wire, it also creates a potential that is perpendicular to the field. The potential decreases as the distance from the wire increases.

What are some real-world applications of the electric potential due to a long wire?

The electric potential due to a long wire is used in various technologies, such as current sensors, antennas, and electromagnets. It is also important in understanding the behavior of electric circuits and the effects of electric fields on conductive materials.

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