Consider a thin plastic rod bent into an arc of radius R and angle a (see figure below). The rod carries a uniformly distributed negative charge -Q. R What are the components E, and E, of the electric field at the origin? Follow the standard four steps. (a) Use a diagram to explain how you will cut up the charged rod, and draw the AF contributed by a representative piece. a (b) Express algebraically the contribution each piece makes to the and y components of the electric field. Be sure to show your integration variable and its origin on your drawing. (Use the following as necessary: Q, R, a, 0, A0, and Eg-) ΔΕ, ΔΕ, - 4ne aR² E₂ Ey @__a sin 6 X 4ne, aR (c) Write the summation as an integral, and simplify the integral as much as possible. State explicitly the range of your integration variable. Lower limit= 0 ✓ de cose X Upper limit= a ✓ Evaluate the integral. (Use the following as necessary: Q, R, a, and EQ.) -sin(a) 4nz aR² Ane aR² (1 cosa) Edit
Consider a thin plastic rod bent into an arc of radius R and angle a (see figure below). The rod carries a uniformly distributed negative charge -Q. R What are the components E, and E, of the electric field at the origin? Follow the standard four steps. (a) Use a diagram to explain how you will cut up the charged rod, and draw the AF contributed by a representative piece. a (b) Express algebraically the contribution each piece makes to the and y components of the electric field. Be sure to show your integration variable and its origin on your drawing. (Use the following as necessary: Q, R, a, 0, A0, and Eg-) ΔΕ, ΔΕ, - 4ne aR² E₂ Ey @__a sin 6 X 4ne, aR (c) Write the summation as an integral, and simplify the integral as much as possible. State explicitly the range of your integration variable. Lower limit= 0 ✓ de cose X Upper limit= a ✓ Evaluate the integral. (Use the following as necessary: Q, R, a, and EQ.) -sin(a) 4nz aR² Ane aR² (1 cosa) Edit
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Solution plz but correctly.
 *(Note: Insert relevant URL of the figure)*
#### Problem Statement
What are the components \(E_x\) and \(E_y\) of the electric field at the origin? Follow the standard four steps:
1. **Use a diagram to explain how you will cut up the charged rod, and draw the \( \Delta \mathbf{E} \) contributed by a representative piece.**
2. **Express algebraically the contribution each piece makes to the \( x \) and \( y \) components of the electric field.**
- Be sure to show your integration variable and its origin on your drawing.
- Use the following as necessary: \( Q \), \( R \), \( \alpha \), \( \theta \), \( \Delta \theta \), and \( \epsilon_0 \).
\[
\Delta E_x = \frac{Q}{4 \pi \epsilon_0 \alpha R^2} (-d\theta \cos \theta)
\]
\[
\Delta E_y = \frac{Q}{4 \pi \epsilon_0 \alpha R^2} (-d\theta \sin \theta)
\]
3. **Write the summation as an integral, and simplify the integral as much as possible.**
- State explicitly the range of your integration variable.
\[
\text{Lower limit: } 0
\]
\[
\text{Upper limit: } \alpha
\]
4. **Evaluate the integral.**
- Use the following as necessary: \( Q \), \( R \), \( \alpha \), and \( \epsilon_0 \).
\[
E_x = \frac{Q}{4 \pi \epsilon_0 \alpha R^2} (-\sin (\alpha))
\]
\[
E](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcbfb1871-07e2-4dce-8005-269462079c4a%2Ff96fd63a-29c0-4441-99f7-b413b4759bd0%2Fc7t804_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Electric Field of a Charged Rod Bent into an Arc
Consider a thin plastic rod bent into an arc of radius \( R \) and angle \( \alpha \) (see figure below). The rod carries a uniformly distributed negative charge \(-Q\).
#### Diagram Description:
A diagram shows a segment of an arc with radius \( R \), subtending an angle \( \alpha \) at the origin of the coordinate system. The rod has a uniformly distributed negative charge \(-Q\).
 *(Note: Insert relevant URL of the figure)*
#### Problem Statement
What are the components \(E_x\) and \(E_y\) of the electric field at the origin? Follow the standard four steps:
1. **Use a diagram to explain how you will cut up the charged rod, and draw the \( \Delta \mathbf{E} \) contributed by a representative piece.**
2. **Express algebraically the contribution each piece makes to the \( x \) and \( y \) components of the electric field.**
- Be sure to show your integration variable and its origin on your drawing.
- Use the following as necessary: \( Q \), \( R \), \( \alpha \), \( \theta \), \( \Delta \theta \), and \( \epsilon_0 \).
\[
\Delta E_x = \frac{Q}{4 \pi \epsilon_0 \alpha R^2} (-d\theta \cos \theta)
\]
\[
\Delta E_y = \frac{Q}{4 \pi \epsilon_0 \alpha R^2} (-d\theta \sin \theta)
\]
3. **Write the summation as an integral, and simplify the integral as much as possible.**
- State explicitly the range of your integration variable.
\[
\text{Lower limit: } 0
\]
\[
\text{Upper limit: } \alpha
\]
4. **Evaluate the integral.**
- Use the following as necessary: \( Q \), \( R \), \( \alpha \), and \( \epsilon_0 \).
\[
E_x = \frac{Q}{4 \pi \epsilon_0 \alpha R^2} (-\sin (\alpha))
\]
\[
E
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