A rod of length L L lies along the y y -axis with its center at the origin. The rod has a nonuniform linear charge density λ = a ∣ y ∣ λ=a|y| , where a is a constant with the units C/m 2 . Draw a graph of λ λ versus y y over the length of the rod.

Hello, fellow physicists today, we're gonna solve the following practice problem together. So first off, let's read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. A bar of length 1.5 L is placed parallel to the X axis. The origin point on the X axis is located at three L divided by four and is measured from the left end of the bar. The charge on the bar varies as lambda equals B multiplied by the absolute value of X where the constant B has the units of coulombs per meter squared sketch a graph of lambda versus X for the total length of the rod. So that is our end goals. We're trying to figure out how we're gonna draw or sketch a graph of lambda versus X considering the total length of the rod. Awesome. So first off, let us assume that the bar is a line of charge with a length. Capital L, we are also told here, we could write that down. So we don't forget. So capital L represents the length. OK. So we are also told that the origin So the origin is at three multiplied by L divided by four on the X axis is measured from the left end of the bar. So the X axis runs from, so this is important. So, so the X axis, it runs from negative three L divided by 42 positive, so negative three L divided by four to positive three L three multiplied by L divided by four. So we're also told that the charge on the bar varies as lambda equals B multiplied by the absolute value of X where B is constant. So like we said before, our end goal is to plot Lambda equals B multiplied by the absolute value of X. And we need to note that the curves will be identical on both the negative and the positive sides. So the greatest value of the absolute value of X is equal to three multiplied by L divided by four. So according to the rules of graphing, lambda depends on X and lambda goes to the Y axis and the length of the bar goes to the X axis. So note that lambda equals B multiplied by the absolute value of X is similar to the straight line equation that states that Y is equal to M multiplied by X plus C. Therefore, our graph will be a straight line. So at this point, we can now put together a table of Lambda and X. Now that we have an idea of the conceptual side of things. So let's note that lambda differs from the absolute value X by the constant B. So I've gone on ahead and I put together a little chart here, what shows how lambda and X V. So how lambda differs from the absolute value of X and this is our table. So as you see when lambda zero, the absolute value of X is zero and when lambda is B multiplied by L divided by four, the absolute value of X is L divided by four and et cetera follows the same pattern. So now we can use the values from our table to help us create our graph of lambda versus X for the total length of the rod. So I've gone on ahead and put together a graph for us. So as you can see when we look at our graph, we know that the X axis runs from like we stated negative three multiplied by L divided by four and it runs to three multiplied by L divided by four. And we also have like from our table, we know that the height like, you know, running on the Y axis, we have the three multiplied by B multiplied by L divided by four and then two negative three multiplied by B multiplied by L divided by four. And that is it. So as you can see, our graph resembles an absolute values sign graph which makes sense. And it has our straight lines here and since we know that the straight line is a. So we know it's similar to the straight line and we'll know that the curves, as we stated will be identical on both the negative, which is on this is the negative. So the left of the graph is negative side to the positive side which is on the right and that's it, that's how you solve for this problem. Hopefully that helped and I can't wait to see you in the next video. Bye.

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Problem 68a

Textbook Question

A rod of length $L$ lies along the $y$ -axis with its center at the origin. The rod has a nonuniform linear charge density $λ = a ∣ y ∣$ , where a is a constant with the units C/m². Draw a graph of $λ$ versus $y$ over the length of the rod.

Verified step by step guidance

Understand the problem: The rod lies along the y-axis, centered at the origin, and has a nonuniform linear charge density λ = a|y|. This means the charge density depends on the absolute value of the y-coordinate, increasing symmetrically as we move away from the origin in either direction along the rod.

Identify the range of y: Since the rod has a length L and is centered at the origin, the y-coordinate ranges from -L/2 to +L/2. This will be the domain of the graph for λ versus y.

Analyze the behavior of λ: The linear charge density λ = a|y| depends on the absolute value of y. For y > 0, λ = ay, and for y < 0, λ = a(-y). This results in a V-shaped graph symmetric about the origin.

Sketch the graph: Plot λ on the vertical axis and y on the horizontal axis. For y = 0 (at the origin), λ = 0. As y increases from 0 to L/2, λ increases linearly as λ = ay. Similarly, as y decreases from 0 to -L/2, λ also increases linearly as λ = a(-y). The graph will have two straight lines meeting at the origin, forming a V-shape.

Label the graph: Mark the endpoints of the graph at y = -L/2 and y = +L/2. At these points, the charge density is λ = a(L/2). Ensure the graph is symmetric about the origin and clearly shows the linear relationship between λ and |y|.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Linear Charge Density

Linear charge density (λ) is defined as the amount of electric charge per unit length along a line. In this case, the charge density varies with the position along the rod, given by the equation λ = a|y|, where 'a' is a constant. Understanding this concept is crucial for analyzing how charge is distributed along the rod and how it affects the electric field around it.

Graphing Functions

Graphing functions involves plotting the relationship between two variables on a coordinate system. For this problem, you will graph λ as a function of y, which requires understanding how to represent the nonuniform charge density visually. This helps in visualizing how the charge density changes with position along the rod, which is essential for further calculations.

Electric Field Due to Charge Distribution

The electric field generated by a charge distribution is influenced by the amount and arrangement of charge. For a rod with a nonuniform charge density, the electric field at a point in space can be calculated by integrating the contributions from each infinitesimal segment of the rod. This concept is vital for understanding the implications of the charge distribution on the surrounding space.

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CALC A 12-cm-long thin rod has the nonuniform charge density $λ (x) = (2.0 nC/cm) e^{- ∣ x ∣ / (6.0 cm)}$ , where x is measured from the center of the rod. What is the total charge on the rod? Hint: This exercise requires an integration. Think about how to handle the absolute value sign.

A rod of length $L$ lies along the $y$ -axis with its center at the origin. The rod has a nonuniform linear charge density $λ=a|y|$ , where a is a constant with the units C/m². Determine the constant a in terms of $L$ and the rod's total charge $Q$ .

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A rod of length LL lies along the yy-axis with its center at the origin. The rod has a nonuniform linear charge density λ=a∣y∣ λ=a|y|, where a is a constant with the units C/m2. Draw a graph of λλ versus yy over the length of the rod.

Key Concepts

Linear Charge Density

Graphing Functions

Electric Field Due to Charge Distribution

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A rod of length $L$ lies along the $y$ -axis with its center at the origin. The rod has a nonuniform linear charge density $λ = a ∣ y ∣$ , where a is a constant with the units C/m². Draw a graph of $λ$ versus $y$ over the length of the rod.