Chapter 23: Problem 27
A uniformly charged, thin ring has radius 15.0 cm and total charge \(+\)24.0 nC. An electron is placed on the ring's axis a distance 30.0 cm from the center of the ring and is constrained to stay on the axis of the ring. The electron is then released from rest. (a) Describe the subsequent motion of the electron. (b) Find the speed of the electron when it reaches the center of the ring.
Short Answer
Step by step solution
Understanding the Problem
Electron's Motion Description
Equipotential Surface Analysis
Calculation of Electric Potential and Energy Conversion
Kinetic Energy at the Center
Calculation of Speed
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Electrostatics
The charged ring creates an electric field in the space surrounding it due to its charge distribution. This field behaves like a map showing the direction and magnitude of the force experienced by a test charge, like our electron, placed in its vicinity.
- Static charges create fields that influence other charges.
- These fields in turn exert electrostatic forces, attracting or repelling charges around them.
Kinetic Energy
The conservation of energy principle suggests that the total energy remains constant. Initially, the electron has maximum potential energy owing to its position relative to the electric potential of the ring. As the electron accelerates towards the center:
- The potential energy decreases because the electron is moving to a lower electric potential.
- This decrease in potential energy results in a corresponding increase in kinetic energy.
Electrostatic Force
Coulomb's Law tells us that the magnitude of the electrostatic force between two point charges is proportional to the product of their charges and inversely proportional to the square of the distance between their centers. For our exercise:
- The electron is negatively charged and the ring is positively charged.
- The electrostatic force is attractive, working to pull the electron towards the ring.
Equipotential Surface
When the electron moves along the axis, it travels through different equipotential levels, causing changes in its potential energy. However, while in complete contact with an equipotential surface, the electron's potential energy remains constant:
- Movement perpendicular to an equipotential surface does not change the potential energy.
- Potential energy changes only when moving through different equipotential surfaces.