1. In the diagram below, a thin plastic rod is charged positively with +Q Coulombs of charge. Applying the principle of superposition of electric fields generated by point particles gives us this integral for the electric field at point P. Explain WHY this is the correct integral. a. What does the term Q/2a represent (in words and a Greek letter symbol). b. What does the numerator in the fraction represent in words and a symbol found on the equation sheet. c. Also draw the numerator on the diagram. Remember, r is the integration variable, not one fixed value.
1. In the diagram below, a thin plastic rod is charged positively with +Q Coulombs of charge. Applying the principle of superposition of electric fields generated by point particles gives us this integral for the electric field at point P. Explain WHY this is the correct integral. a. What does the term Q/2a represent (in words and a Greek letter symbol). b. What does the numerator in the fraction represent in words and a symbol found on the equation sheet. c. Also draw the numerator on the diagram. Remember, r is the integration variable, not one fixed value.
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![1. In the diagram below, a thin plastic rod is charged positively with \( +Q \) Coulombs of charge. Applying the principle of superposition of electric fields generated by point particles gives us this integral for the electric field at point \( P \). Explain WHY this is the correct integral.
a. What does the term \( Q/2a \) represent (in words and a Greek letter symbol).
b. What does the numerator in the fraction represent in words and a symbol found on the equation sheet.
c. Also draw the numerator on the diagram. Remember, \( x \) is the integration variable, not one fixed value.
\[
\vec{E} = k_e \int_{x=-a}^{x=a} \frac{Q}{2a} \left( \frac{y \hat{j} - x \hat{i}}{(x^2 + y^2)^{3/2}} \right) dx
\]
**Diagram Explanation:**
- The diagram consists of a horizontal thin rod aligned along the x-axis, with its ends labeled as \( -a \) and \( +a \).
- A point \( P \) is located above the center of the rod (on the y-axis).
- The integral expression provided calculates the electric field \( \vec{E} \) at point \( P \) due to the charge distribution on the rod.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa06fb91f-1445-4bd6-b9dd-268642fc0d22%2F73b17685-2a09-4c50-8e75-5c6bbca8156c%2Fwttqd9a_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1. In the diagram below, a thin plastic rod is charged positively with \( +Q \) Coulombs of charge. Applying the principle of superposition of electric fields generated by point particles gives us this integral for the electric field at point \( P \). Explain WHY this is the correct integral.
a. What does the term \( Q/2a \) represent (in words and a Greek letter symbol).
b. What does the numerator in the fraction represent in words and a symbol found on the equation sheet.
c. Also draw the numerator on the diagram. Remember, \( x \) is the integration variable, not one fixed value.
\[
\vec{E} = k_e \int_{x=-a}^{x=a} \frac{Q}{2a} \left( \frac{y \hat{j} - x \hat{i}}{(x^2 + y^2)^{3/2}} \right) dx
\]
**Diagram Explanation:**
- The diagram consists of a horizontal thin rod aligned along the x-axis, with its ends labeled as \( -a \) and \( +a \).
- A point \( P \) is located above the center of the rod (on the y-axis).
- The integral expression provided calculates the electric field \( \vec{E} \) at point \( P \) due to the charge distribution on the rod.
Expert Solution
Step 1
Solution:
1). The principle of superposition gives the total electric field at point p due to the charged plastic rod. The charge given to the rod of length 2a is Q. Let us consider a small charge element dq of width dx at a distance x from the origin. Also, let y be the distance of point p from the center of the rod. Then, the electric field at point p due to charge dq can be written as the following,
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