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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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### 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 \(-Q\). ![Diagram](URL) *(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
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 \(-Q\). ![Diagram](URL) *(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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