Chapter 23: Problem 50
A small sphere with mass 5.00 \(\times 10^{-7}\) kg and charge \(+\)7.00 \(\mu\)C is released from rest a distance of 0.400 m above a large horizontal insulating sheet of charge that has uniform surface charge density \(\sigma = +\)8.00 pC\(/\)m\(^2\). Using energy methods, calculate the speed of the sphere when it is 0.100 m above the sheet.
Short Answer
Step by step solution
Determine Change in Electric Potential Energy
Calculate the Electric Field
Compute Change in Electric Potential
Determine Change in Potential Energy
Apply Energy Conservation
Solve for Speed
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Electric Field
To find the strength of the electric field (\( E \)) generated by such a sheet, we use the formula:
- \[ E = \frac{\sigma}{2\varepsilon_0} \]
This formula tells us that the electric field strength depends on how much charge is spread across a given area of the sheet. The key point is that the field created by an infinite sheet is constant and not dependent on the distance from the sheet.
This is why the current scenario uses the electric field calculation to determine potential differences and energy changes, crucial for finding the sphere's speed.
Surface Charge Density
In this problem, the surface charge density is given as \( +8.00 \text{ pC/m}^2 \). This positive value indicates that the sheet is positively charged, resulting in an outward electric field.
Calculating \( \sigma \) is crucial because it directly influences the electric field produced by the sheet. Recall, a higher surface charge density produces a stronger electric field:
- If the charge density were higher, the electric field strength would increase.
- Conversely, a lower charge density would decrease the field strength.
Energy Conservation
In this exercise, the small sphere's potential energy decreases as it moves closer to the charged sheet. Here's how it works:
- Initially, the sphere has a potential energy due to its position in the electric field.
- As it moves towards the charged sheet, potential energy \( \Delta U \) decreases.
Overall, this transformation characterizes how energy conservation governs the dynamics of charged objects in electric fields.