Answer :
To address the question, we need to analyze the given options for the potential difference when moving a proton from a position at "six o'clock" to "ten o'clock" on the clock face in the [tex]\( x-y \)[/tex] plane, with respect to a disk of positive charge [tex]\( Q \)[/tex].
### Understanding the given:
1. Initial Position ("six o'clock"): The proton is at a distance [tex]\( R \)[/tex] from the center of the disk, directly below at the positive y-axis.
2. Final Position ("ten o'clock"): The proton should be moved to [tex]\( R/2 \)[/tex] from the center, making an angle corresponding to "ten o'clock" with respect to the [tex]\( x-y \)[/tex] plane.
### Constants:
- [tex]\( Q \)[/tex] is the charge.
- [tex]\( k_c \)[/tex] is Coulomb's constant, which has a value of approximately [tex]\( 8.99 \times 10^9 \, \text{N·m}^2/\text{C}^2 \)[/tex].
- [tex]\( R \)[/tex] is the initial distance from the center of the disk to the proton's position at "six o'clock".
### Calculating the Potential Differences:
Let's analyze the potential difference expressions one by one:
1. Option (A): [tex]\( k_c Q\left(\frac{1}{R}\right) \)[/tex]
- This implies a potential difference solely dependent on the distance [tex]\( R \)[/tex].
2. Option (B): [tex]\( k_c Q q\left(\frac{1}{R}\right) \)[/tex]
- Here, [tex]\( q \)[/tex] (usually representing a second charge) multiplies the term, which does not fit our problem description where we only have a single moving proton.
3. Option (C): [tex]\( k_c Q\left(\frac{1}{R}\right)\left(\frac{2 \pi}{3}\right) \)[/tex]
- This suggests a potential difference considering an angular displacement (in this case, [tex]\( \frac{2 \pi}{3} \)[/tex] radians or 120 degrees), which fits the change from "six o'clock" to "ten o'clock".
4. Option (D): [tex]\( -k_c Q\left(\frac{1}{R}\right) \)[/tex]
- This represents a potential difference inversely related to option (A), with a negative sign, suggesting a decrease, which might be relevant if the potential at the end position is lower.
5. Option (E): [tex]\( k_c Q q\left(\frac{1}{R^2}\right) \)[/tex]
- Again includes a second charge [tex]\( q \)[/tex], and an inverse square dependency, not fitting our scenario exactly.
### Matching Numerical Results:
When compared to the numerical results:
- Option (A), (B), (D) all yield the value [tex]\( 8.99 \times 10^9 \, \text{V} \)[/tex], which means they give equivalent potential differences when plugged into the problem's context.
- Option (C) yields [tex]\( 18.8286 \times 10^9 \, \text{V} \)[/tex], which accounts for the geometry of the motion (moving across a specific angle), this matches a more elaborate geometric consideration.
- Option (E) has a differing structure as it involves [tex]\( R^2 \)[/tex], returning similar potential difference to Option (B) but squared term dependent.
Thus, the most appropriate expression for the potential difference, taking into account the movement from "six o'clock" to "ten o'clock", is:
(C) [tex]\( k_c Q\left(\frac{1}{R}\right)\left(\frac{2 \pi}{3}\right) \)[/tex]
This considers the geometrical path and provides the expected numerical match suitable for the setup provided.
### Understanding the given:
1. Initial Position ("six o'clock"): The proton is at a distance [tex]\( R \)[/tex] from the center of the disk, directly below at the positive y-axis.
2. Final Position ("ten o'clock"): The proton should be moved to [tex]\( R/2 \)[/tex] from the center, making an angle corresponding to "ten o'clock" with respect to the [tex]\( x-y \)[/tex] plane.
### Constants:
- [tex]\( Q \)[/tex] is the charge.
- [tex]\( k_c \)[/tex] is Coulomb's constant, which has a value of approximately [tex]\( 8.99 \times 10^9 \, \text{N·m}^2/\text{C}^2 \)[/tex].
- [tex]\( R \)[/tex] is the initial distance from the center of the disk to the proton's position at "six o'clock".
### Calculating the Potential Differences:
Let's analyze the potential difference expressions one by one:
1. Option (A): [tex]\( k_c Q\left(\frac{1}{R}\right) \)[/tex]
- This implies a potential difference solely dependent on the distance [tex]\( R \)[/tex].
2. Option (B): [tex]\( k_c Q q\left(\frac{1}{R}\right) \)[/tex]
- Here, [tex]\( q \)[/tex] (usually representing a second charge) multiplies the term, which does not fit our problem description where we only have a single moving proton.
3. Option (C): [tex]\( k_c Q\left(\frac{1}{R}\right)\left(\frac{2 \pi}{3}\right) \)[/tex]
- This suggests a potential difference considering an angular displacement (in this case, [tex]\( \frac{2 \pi}{3} \)[/tex] radians or 120 degrees), which fits the change from "six o'clock" to "ten o'clock".
4. Option (D): [tex]\( -k_c Q\left(\frac{1}{R}\right) \)[/tex]
- This represents a potential difference inversely related to option (A), with a negative sign, suggesting a decrease, which might be relevant if the potential at the end position is lower.
5. Option (E): [tex]\( k_c Q q\left(\frac{1}{R^2}\right) \)[/tex]
- Again includes a second charge [tex]\( q \)[/tex], and an inverse square dependency, not fitting our scenario exactly.
### Matching Numerical Results:
When compared to the numerical results:
- Option (A), (B), (D) all yield the value [tex]\( 8.99 \times 10^9 \, \text{V} \)[/tex], which means they give equivalent potential differences when plugged into the problem's context.
- Option (C) yields [tex]\( 18.8286 \times 10^9 \, \text{V} \)[/tex], which accounts for the geometry of the motion (moving across a specific angle), this matches a more elaborate geometric consideration.
- Option (E) has a differing structure as it involves [tex]\( R^2 \)[/tex], returning similar potential difference to Option (B) but squared term dependent.
Thus, the most appropriate expression for the potential difference, taking into account the movement from "six o'clock" to "ten o'clock", is:
(C) [tex]\( k_c Q\left(\frac{1}{R}\right)\left(\frac{2 \pi}{3}\right) \)[/tex]
This considers the geometrical path and provides the expected numerical match suitable for the setup provided.
