A plastic rod has been bent into a circle of radius R = 8.80 cm. It has a

charge Q1 = +4.60 pC uniformly distributed along one-quarter of its circumference and a
charge Q2 = -6Q1 uniformly distributed along the rest of the circumference. Take V = 0 at
infinity.

(a) What is the electric potential at the center C of the circle?

V

(b) What is the electric potential at point P, which is on the central axis of the circle at

distance D = 6.71 cm from the center?

V

To find the electric potential at the center C of the circle, we can consider the contributions from the charges Q1 and Q2 separately.

(a) Electric potential at the center C:

1. Calculate the electric potential due to the charge Q1:
- The charge Q1 is distributed along one-quarter of the circumference of the circle. Since the entire circumference is 2πR, the length of the quarter circumference is (1/4)(2πR).
- To find the electric potential due to charge Q1 at the center, we can use the formula for the electric potential due to a charged rod: V1 = k * (Q1 / r1), where k is the Coulomb's constant, Q1 is the charge, and r1 is the distance from Q1 to the center C.
- In this case, the distance r1 is equal to the radius of the circle R.
- Substitute the values into the formula: V1 = k * (Q1 / R).

2. Calculate the electric potential due to the charge Q2:
- The charge Q2 is distributed along three-quarters of the circumference of the circle. Since the entire circumference is 2πR, the length of these three-quarters is (3/4)(2πR).
- To find the electric potential due to charge Q2 at the center, we can again use the formula for the electric potential due to a charged rod: V2 = k * (Q2 / r2), where Q2 is the charge, and r2 is the distance from Q2 to the center C.
- In this case, the distance r2 is equal to the radius of the circle R.
- Substitute the values into the formula: V2 = k * (Q2 / R).

3. Add up the contributions of V1 and V2 to find the total electric potential at the center: V = V1 + V2.

(b) Electric potential at point P:

To find the electric potential at point P on the central axis of the circle, we can use a similar approach.

1. Calculate the electric potential due to the charge Q1:
- In this case, the distance from Q1 to point P is given as D.
- Use the formula for the electric potential: V1 = k * (Q1 / r1), where Q1 is the charge, and r1 is the distance from Q1 to point P.
- In this case, the distance r1 is equal to D, the given distance.
- Substitute the values into the formula: V1 = k * (Q1 / D).

2. Calculate the electric potential due to the charge Q2:
- In this case, the distance from Q2 to point P is also D, as it is located on the central axis.
- Use the formula for the electric potential: V2 = k * (Q2 / r2), where Q2 is the charge, and r2 is the distance from Q2 to point P.
- In this case, the distance r2 is equal to D, the given distance.
- Substitute the values into the formula: V2 = k * (Q2 / D).

3. Add up the contributions of V1 and V2 to find the total electric potential at point P: V = V1 + V2.

Remember to replace the value of k with the appropriate constant value, Q1 and Q2 with their numerical values, R with the given radius, and D with the given distance. Finally, calculate the electric potential using the formulas provided.