A thin rod of length L and total charge Q has the nonuniform l...

Question

A thin rod of length L and total charge Q has the nonuniform linear charge distribution λ(x)=λ₀x/L, where x is measured from the rod's left end. What is the electric potential on the axis at distance d left of the rod's left end?

Accepted Answer

Hello, let's go through this practice problem. A charged plastic rod of length L has a linear charge density given by lambda of X equals lambda not multiplied by one minus X divided by L along its length where X is measured from one end of the rod, determine the electric potential at a point P located on the axis at a distance 10 L from the center of the rod. Option A one divided by 10 L multiplied by K Lambda knot multiplied by one minus X divided by L. Option BK Lambda knot multiplied by 10 multiplied by the natural log of negative 21 L divided by two. Option CK Lambda knot multiplied by 1.1001 plus the natural log of negative 19 L divided by two minus the natural log of negative 21 L divided by two. And option D is the exact same as option C except the natural log terms are multiplied by 10. So we had a rod and this rod has a length of 10 or is the length of just L but at a length of 10 L away from the center of this rod is a point P. So I'll label the center of the rod and there is a distance of 10 L between that central point and point B which is drawn on the right side. We say that this rod rests along the X axis. And we're looking to find the electric potential at point P. Now recall that electric potential V is equal to the Coolum constant K multiplied by the charge of whatever is setting up the potential divided by the distance between that point charge and the point we're looking at this is the basic formula for the electric potential due to a point charge. But it's a little more complicated in this case because we're not dealing with a point charge, we're dealing with a length of charge that is along the same axis as the point. So this is going to become a calculus problem where instead of just taking the electric potential due to a point, we're going to integrate the infinitesimal or the differential potential over the charges. So our V equation becomes AD V equation and the Q for charge becomes ad Q because we're going to be integrating individual elements of the charge of the rod over the rod's length. So let's say that we're integrating tiny strips segments of the rod and it'll say that each one of these elements has a charge of DQ. And the length of distance between DQ and the point P is what we'll is what we will be calling R. So to find the potential, we're going to integrate DV. So we're integrating K DQ divided by R. So we're essentially integrating K divided by R with respect to Q, the K is a constant. However, so we'll move that outside the integral. And since we don't actually know that much about Q, it's going to be more convenient for us to instead of using DQ instead to rewrite this in terms of the length of the rod. So recall that another way to represent charge is that charge is equal to charge density multiplied by length or in the context of our integral DQ, the element of charge is equal to the charge density multiplied by the element of length. So that means that our integral is equal to K multiplied by the integral of LAMBDA DX divided by R. So now this is an integral with respect to our position along the X axis. And since we want to integrate a charge over a rod, our bounds of integration should be what will give us the length of the rod. So I'll set our origin point to be right at the center of the rod, which means that the hour of bounds of integration will be from the left side of the rod. So negative half of the length up to positive half of the length. And we're integrating the linear charge density lambda, which is actually given to us in the problem as a formula where the problem directly tells us that Lambda is equal to Lambda, not multiplied by one minus X divided by L. And this is all multiplied by DX divided by R. And recall that R is the distance between point B and each individual point on the rod that we're looking at. But since the distance between point P and the center of the rod is given as 10 L, this means that R can actually be represented as 10 L minus X. Because if you take the distance between P and the center of the rod and then subtract whatever point on the rod, we're looking at lengthwise, that's gonna give us the equivalent of R. So we're just narrowing down the variables that we have in the equation anyways lambda knot is also a constant. So we're gonna pull that out of the integral. So we're integrating from negative L divided by two to positive L divided by two. It'll distribute the DX divided by 10 L minus X over the parentheses. So it becomes DX divided by 10 L minus X minus XDX divided by 10 L squared minus LX. And because we've now split the DX into two different terms, we actually kind of have two integrals here that we can solve. And when we perform this integral, we find that it's equal to K multiplied by Lambda knot from the uh multiplied by 0.1001 minus 10, multiplied by the natural log of negative 21 L divided by two minus 10, multiplied by the natural log of a negative 19 L divided by two minus one. And there's of course, some simplification we can do here, this minus one at the end can add on to the 0.1001 at the very beginning because the negative signs cancel out. And in general will just distribute the negative signs through the parentheses. And so this gives us a new value if K multiplied by lambda knot multiplied by 1.1001 plus 10, multiplied by the natural log of negative 19 L divided by two minus 10 multiplied by the natural log of negative 21 L divided by two. And that is the answer to this problem which you'll notice corresponds to option D. So option D is our answer to this problem. And that is it for this problem. I hope this video helped you out and please consider checking out our other tutoring videos which will give you more experience with these types of problems. That's all for now. And I hope you have a lovely day. Bye bye.

A thin rod of length L and total charge Q has the nonuniform linear charge distribution λ(x)=λ₀x/L, where x is measured from the rod's left end. What is the electric potential on the axis at distance d left of the rod's left end?

Key Concepts

Electric Potential

Linear Charge Density

Integration in Electric Field Calculations

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