**Example: The infinitely thin, uniformly charged line of finite length. Below shows the line in yellow:**This example illustrates a uniformly charged, infinitely thin line of finite length.**Diagram Explanation:**- The diagram depicts a coordinate system with the `x` and `y` axes labeled.- A yellow segment represents the charged line, positioned along the `x`-axis, extending from `x = a` to `x = a + l`, where `a` is the distance from the origin to the start of the line, and `l` is the length of the line.- A point `P` is marked at the origin (0,0).- A differential element of the line, denoted as `dx`, is highlighted within the yellow line.- The charge of the line segment `dx` is represented by `dq = λdx`, where `λ` is the linear charge density (charge per unit length).**Description:**It is uniformly charged, that is, none of the charge is bunched up to the left or right of the line (like a row of protons side by side). The line has length `l` and is a distance `a` from the origin. It has charge `Q`. What is the electric field at the origin? **Non-Uniform Line**Suppose that the charged line in figure 6.20 is non-uniformly charged such that it has a linear charge density of \( \lambda = \frac{C}{x} \) where \( C \) is a constant. Notice that more charge is located near the left side of the line (and is not evenly spread out).What is the correct expression for the electric field at the origin (figure 6.20) with this charge density? (hint: substitute \( \lambda \) into the integral and integrate).Select an answer and submit. For keyboard navigation, use the up/down arrow keys to select an answer.a) \( kCl \)b) \( \frac{1}{2}kC(l^2 + 2la) \)c) \( kCln \left( \frac{l + a}{a} \right) \)d) \( \frac{kC}{l + a} \)e) \( \frac{1}{2}kC \left( \frac{1}{a^2} - \frac{1}{(l + a)^2} \right) \)

Electric field due to non-linear charge is calculated by E=∫kdqx2 Where, dq is the charge of the…

Non-Uniform Line -- Suppose that the charged line in figure 6.20 is non-uniformly charged such that it has a linear charge density C of = where C is a constant. Notice that more charge is located near the left side of the line (and is not evenly spread out). X What is the correct expression for the electric field at the origin (figure 6.20) with this charge density? (hint: substitute λ into the integral and integrate). Select an answer and submit. For keyboard navigation, use the up/down arrow keys to select an answer. a kCl b kC(1² + 2la) ckCin(+a) с d kC l+a 1 1/2 KC(=/²2 - (1 + a)²2) 1 e

Non-Uniform Line -- Suppose that the charged line in figure 6.20 is non-uniformly charged such that it has a linear charge density C of = where C is a constant. Notice that more charge is located near the left side of the line (and is not evenly spread out). X What is the correct expression for the electric field at the origin (figure 6.20) with this charge density? (hint: substitute λ into the integral and integrate). Select an answer and submit. For keyboard navigation, use the up/down arrow keys to select an answer. a kCl b kC(1² + 2la) ckCin(+a) с d kC l+a 1 1/2 KC(=/²2 - (1 + a)²2) 1 e

Principles of Physics: A Calculus-Based Text

5th Edition

ISBN:9781133104261

Author:Raymond A. Serway, John W. Jewett

Publisher:Raymond A. Serway, John W. Jewett

Chapter20: Electric Potential And Capacitance

Section: Chapter Questions

Problem 14P: Review. A light, unstressed spring has length d. Two identical particles, each with charge q, are...

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Concept explainers

Electric Charges and Fields

One of the fundamental properties is the electromagnetic property. Charge cannot be destroyed by any process and this contributes formally to the law of charge conservation.

Question

**Example: The infinitely thin, uniformly charged line of finite length. Below shows the line in yellow:**

This example illustrates a uniformly charged, infinitely thin line of finite length.

**Diagram Explanation:**

- The diagram depicts a coordinate system with the `x` and `y` axes labeled.
- A yellow segment represents the charged line, positioned along the `x`-axis, extending from `x = a` to `x = a + l`, where `a` is the distance from the origin to the start of the line, and `l` is the length of the line.
- A point `P` is marked at the origin (0,0).
- A differential element of the line, denoted as `dx`, is highlighted within the yellow line.
- The charge of the line segment `dx` is represented by `dq = λdx`, where `λ` is the linear charge density (charge per unit length).

**Description:**

It is uniformly charged, that is, none of the charge is bunched up to the left or right of the line (like a row of protons side by side). The line has length `l` and is a distance `a` from the origin. It has charge `Q`. What is the electric field at the origin?

**Non-Uniform Line**

Suppose that the charged line in figure 6.20 is non-uniformly charged such that it has a linear charge density of \( \lambda = \frac{C}{x} \) where \( C \) is a constant. Notice that more charge is located near the left side of the line (and is not evenly spread out).

What is the correct expression for the electric field at the origin (figure 6.20) with this charge density? (hint: substitute \( \lambda \) into the integral and integrate).

Select an answer and submit. For keyboard navigation, use the up/down arrow keys to select an answer.

a) \( kCl \)

b) \( \frac{1}{2}kC(l^2 + 2la) \)

c) \( kCln \left( \frac{l + a}{a} \right) \)

d) \( \frac{kC}{l + a} \)

e) \( \frac{1}{2}kC \left( \frac{1}{a^2} - \frac{1}{(l + a)^2} \right) \)

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