How Do You Calculate Electric Potential and Field for a Finite Line Charge?

In summary, the conversation discusses finding the electric potential at a point on the x-axis due to a line charge with density λ extending from -a to +a. The solution involves using the equation V = \frac{1}{4\pi \epsilon _0}\int\frac{dq}{r} and making a substitution, leading to the result \frac{Q}{4\pi \epsilon _0 r}. However, there are some uncertainties about the integration and the presence of λ in the answer. It is suggested to also consider using Gauss's law and the equation E=-dV/dr to find the electric field.
  • #1
Brewer
212
0

Homework Statement


A line charge has density λ and extends along the x-axis from -a to +a. Find the electric potential at a point r on the x-axis (r>a). Use your result to find the E-field at r.

Homework Equations


[tex]V = \frac{1}{4\pi \epsilon _0}\int\frac{dq}{r} [/tex]

The Attempt at a Solution


I've said so far:
dQ = λdx = Qdx/2a

Then I made the substitution into the above equation, and integrated wrt x with limits ±a, leaving me with [tex]\frac{Q}{4\pi \epsilon _0 r}[/tex].

However I'm not sure I made the correct integration here. I also believe that the answer for V should have λ in it somewhere. Have I gone wrong somewhere?

I haven't actually gotten round to looking at the E-field yet - I'll get there once this is completed!

Any pointers would be appreciated.

Thanks
 
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  • #2
Is this correct? I've spent my morning researching this, and I can't seem to find anything to compare it to - I'm a little confused as to what to do when the point is somewhere on the same line. Are my limits correct?
 
  • #3
You could find the Electric field using gauss's law and then use E=-dV/dr to solve for V.
 

FAQ: How Do You Calculate Electric Potential and Field for a Finite Line Charge?

What is a line charge?

A line charge is a theoretical construct used in electrostatics to represent a one-dimensional distribution of electric charge. It is an idealization that is often used to simplify calculations and understand the behavior of continuous charge distributions.

How is the electric field of a line charge calculated?

The electric field of a line charge can be calculated using Coulomb's law, which states that the electric field at a point is directly proportional to the magnitude of the charge and inversely proportional to the square of the distance from the point to the charge. The electric field of a line charge can also be calculated using Gauss's law, which relates the electric field to the charge enclosed by a closed surface.

What is the difference between a positive and negative line charge?

A positive line charge has a net positive charge along its length, meaning that the charge is distributed in a way that creates an outward electric field. A negative line charge has a net negative charge along its length, resulting in an inward electric field. The direction of the electric field is determined by the sign of the charge: positive charges create outward electric fields, while negative charges create inward electric fields.

How does the electric field of a line charge vary with distance?

The electric field of a line charge varies inversely with distance from the charge. As the distance from the charge increases, the strength of the electric field decreases. This relationship is described by the inverse square law, which states that the electric field is inversely proportional to the square of the distance from the charge.

What are some real-life examples of line charges?

Line charges are not found in nature, but they can be used to model and understand the behavior of real-life systems. For example, a thin wire carrying a constant electric charge can be approximated as a line charge. Additionally, a charged particle moving along a straight path can also be modeled as a line charge. Line charges are also used to explain the behavior of lightning and electric discharges.

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