Chapter 22: Problem 32
A very small object with mass 8.20 \(\times\) 10\(^{-9}\) kg and positive charge 6.50 \(\times\) 10\(^{-9}\) C is projected directly toward a very large insulating sheet of positive charge that has uniform surface charge density 5.90 \(\times\) 10\(^{-8}\) C/m2. The object is initially 0.400 m from the sheet. What initial speed must the object have in order for its closest distance of approach to the sheet to be 0.100 m?
Short Answer
Step by step solution
Understanding the Problem
Using Energy Conservation
Writing the Energy Equation
Electric Field from the Sheet
Calculating Initial Potential Energy
Calculating Potential Energy at Closest Approach
Energy Conservation Equation
Solve for Initial Speed
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Energy Conservation
Initially, the object possesses kinetic energy, which depends on its mass and speed, described by the formula: \[ KE = \frac{1}{2}mv^2 \] where \( m \) is the mass and \( v \) is the speed.
The object also has electric potential energy due to its position relative to the charged sheet. The potential energy \( U \) is given by: \[ U = qEx \] where \( q \) is the charge, \( E \) is the electric field, and \( x \) is the distance to the sheet.
When the charged object approaches the sheet and stops momentarily, it converts its initial kinetic energy into potential energy. This scenario exemplifies energy conservation, where the initial sum of kinetic and potential energy equals the energy at the closest approach point, which is only potential energy at that moment. By balancing these energies, we can solve for the initial speed of the object.
Electric Field
This setup creates a uniform electric field, which simplifies calculations by making \( E \) constant no matter the distance as long as it stays close and parallel to the sheet. In this exercise, \( E \) determines the change in potential energy as the charged object moves towards the sheet. The electric field's strength influences the object's acceleration, illustrating how electric fields govern motion and energy in electrostatic contexts.
Surface Charge Density
The value of \( \sigma \) affects the electric field produced by the sheet according to the formula: \[ E = \frac{\sigma}{2\varepsilon_0} \] where \( E \) is the resultant electric field due to the charged sheet, allowing us to compute the electric potential energy when a charged object is nearby.
High surface charge density results in stronger electric fields, leading to larger potential energy changes for charged particles within its influence. Thus, knowing \( \sigma \) is vital when evaluating how charged objects will behave around charged surfaces, making it a critical factor in both calculating forces and the resulting energies involved in such systems. This in turn ties into our energy conservation calculations as we determine energy relations with respect to distances from the charged sheet.