A 2.0-cm-diameter copper ring has 5.0×10⁹ excess electrons. A proton is released from rest on the axis of the ring, 5.0 cm from its center. What is the proton's speed as it passes through the center of the ring?

Verified step by step guidance

1. Determine the total charge on the copper ring due to the excess electrons. Use the formula for charge: \( Q = n \cdot e \), where \( n \) is the number of excess electrons and \( e \) is the elementary charge (\( e = 1.6 \times 10^{-19} \; \text{C} \)).
2. Calculate the electric potential at the center of the ring due to the charge on the ring. The formula for the electric potential at the center of a ring is \( V = \frac{k \cdot Q}{R} \), where \( k \) is Coulomb's constant (\( k = 8.99 \times 10^9 \; \text{N·m}^2/\text{C}^2 \)) and \( R \) is the radius of the ring (half the diameter).
3. Determine the initial electric potential energy of the proton when it is 5.0 cm away from the center of the ring. Use the formula \( U = q \cdot V \), where \( q \) is the charge of the proton (\( q = 1.6 \times 10^{-19} \; \text{C} \)) and \( V \) is the electric potential at that point. The potential at a distance \( x \) from the center of the ring can be calculated using \( V = \frac{k \cdot Q}{\sqrt{R^2 + x^2}} \).
4. Apply the principle of conservation of energy. The initial electric potential energy of the proton is converted into its kinetic energy as it passes through the center of the ring. Use the equation \( U_{\text{initial}} = K_{\text{final}} \), where \( K_{\text{final}} = \frac{1}{2} m v^2 \), \( m \) is the mass of the proton (\( m = 1.67 \times 10^{-27} \; \text{kg} \)), and \( v \) is the speed of the proton.
5. Solve for the speed \( v \) of the proton by rearranging the kinetic energy equation: \( v = \sqrt{\frac{2 \cdot U_{\text{initial}}}{m}} \). Substitute the values for \( U_{\text{initial}} \) and \( m \) to find the proton's speed as it passes through the center of the ring.

Key Concepts

Electric Field

The electric field is a vector field surrounding charged particles that exerts a force on other charged objects. It is defined as the force per unit charge experienced by a positive test charge placed in the field. In this scenario, the excess electrons in the copper ring create an electric field that influences the motion of the proton as it moves towards the center.

Coulomb's Law

Coulomb's Law describes the force between two point charges. It states that the force is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. This law is essential for calculating the force acting on the proton due to the electric field created by the excess electrons in the ring.

Conservation of Energy

The principle of conservation of energy states that energy cannot be created or destroyed, only transformed from one form to another. In this problem, the potential energy of the proton in the electric field is converted into kinetic energy as it accelerates towards the center of the ring. This relationship allows us to calculate the proton's speed as it passes through the center.