# Electric Potential, Linear charge with a loop

In summary, the electric potential at point O, for a wire of finite length with a uniform linear charge density \lambda bent into a specific shape, can be found using the equation k_{e} \lambda (\pi + 2ln3). However, there may be a typo in the original post as the correct answer should only have one pi. The solution involves breaking the wire into three pieces (two straight segments and one curved) and setting up the appropriate integrations for each piece.
1. A wire of finite length that has a uniform linear charge density $$\lambda$$ is bent into the shape shown in the figure below. Find the electric potential at point O.
The image has the setup.

2. The answer is $$k\lambda\pi (\pi + 2ln3)$$ How the hell do I get this? The primary equation I am using is of course $$\int \frac{dq}{r}$$ where finding a proper dq is the chore.

3. Okay, so this problem is driving me a little bit crazy. I tried integrating the electric potential equation with respect to the loop, integrating from 0 to $$\pi$$ which gives me $$k\lambda\pi$$
It makes sense that this is incorrect as the the linear charge density is going to be spread across the entire wire. However, whenever I try to account for the rest of the wire outside of the loop I get nonsensical and incorrect answers and integrations. I am fairly certain this is what I need to do, I am just clueless as to how to go about it
How do I set this up ? Thanks for any help you can provide.

fixing latex if it something looks funny

#### Attachments

• wire.bmp
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Treat this as three pieces: Two straight segments and one curved. The trick to the curved piece is that it's part of a circle. (No integration needed for that part!)

Looks like you got the curved part, now set up the integration for the straight segments.

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I am having big hangups with the straight pieces and what I am supposed to do with them after I have found them. Already knowing the answer is further messing me up because I know there is no R in there and my functions for the straight pieces spit out natural logs of R and stuff.

$$\int \frac {k \lambda dx}{x}$$ I don't know how much sense this makes but I was trying to find the potential at 0 in reference to a piece of dx on the right hand segment where it starts at R and ends at 3R.

that is supposed to be marked as Integral from R to 3R but I could not find the proper latex syntax to display it correctly.

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You left out a factor of k, and the limits should be from R to 3R, but otherwise your integral is OK. Hint: $\ln (a) - \ln (b) = \ln (a/b)$.

FYI:
$$\int_{R}^{3R} \frac{k \lambda}{x}dx$$

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I didn't leave out the k, my poor latex omitted the k when i had the incorrect syntax for the integral from R to 3R. So then with both sides I have $$k \lambda 2ln3$$ and then the loop makes $$k\lambda\pi$$ Where is that extra PI coming from in the answer ? If I add these it will be missing...

$$k_{e} \lambda (\pi + 2ln3)$$ If I integrate this from 0 to PI with respect to dtheta I would get the extra PI but why would I do that?

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i just did the same question.
i don't know where that first one came from.

So then with both sides I have $$k \lambda 2ln3$$ and then the loop makes $$k\lambda\pi$$ Where is that extra PI coming from in the answer ? If I add these it will be missing...

$$k_{e} \lambda (\pi + 2ln3)$$
This answer is correct. The extra pi in the original post is a typo. (I must not have seen this last post. Oops. )

## 1. What is electric potential?

Electric potential is a measure of the potential energy per unit charge at a given point in an electric field. It is measured in volts (V).

## 2. How is electric potential calculated?

Electric potential is calculated by dividing the electric potential energy by the charge at a given point in an electric field. It can also be calculated by integrating the electric field over a certain distance.

## 3. What is linear charge?

Linear charge refers to a charged object that has a uniform distribution of charge along its length. It is typically represented by the symbol λ and is measured in coulombs per meter (C/m).

## 4. What is a loop in terms of electric potential?

A loop in terms of electric potential refers to a closed path or circuit in which the total change in electric potential is zero. This means that the starting and ending points have the same electric potential.

## 5. How does a loop affect electric potential?

A loop can affect electric potential by creating a magnetic field, which in turn can affect the electric potential at different points along the loop. This is known as electromagnetic induction and is the principle behind generators and transformers.

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