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(II) A person scuffing her feet on a wool rug on a dry day accumulates a net charge of \(-46 \mu\) C. How many excess electrons does she get, and by how much does her mass increase?

Short Answer

Expert verified
She gains approximately \(2.875 \times 10^{14}\) excess electrons and her mass increases by \(2.618 \times 10^{-16}\, kg\).

Step by step solution

01

Understanding the Charge

The problem states a net charge of \(-46 \mu C\). This represents an excess of negative charge due to additional electrons. Since the charge of one electron is \(-1.6 \times 10^{-19} C\), we will use this information to find the number of excess electrons.
02

Finding Excess Electrons

To find the number of excess electrons, we divide the total charge by the charge of one electron: \[ n = \frac{-46 \times 10^{-6}\, C}{-1.6 \times 10^{-19}\, C/e^-} \] This calculation gives us the number of electrons gained by scuffing feet.
03

Calculating Excess Electrons

Compute the number of excess electrons: \[ n = \frac{46 \times 10^{-6}}{1.6 \times 10^{-19}} = 2.875 \times 10^{14} \] Thus, she gains approximately \(2.875 \times 10^{14}\) excess electrons.
04

Understanding Mass Change

Now, we need to find how much the person's mass increases due to these excess electrons. The mass of one electron is approximately \(9.11 \times 10^{-31}\, kg\).
05

Calculating Mass Change

Find the total mass of the excess electrons by multiplying the number of excess electrons by the mass of one electron: \[ \Delta m = 2.875 \times 10^{14} \times 9.11 \times 10^{-31}\, kg = 2.618 \times 10^{-16}\, kg \] Therefore, the person's mass increases by \(2.618 \times 10^{-16}\, kg\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Charge of an Electron
When we talk about the charge of an electron, we're referring to a fundamental property of electrons that contributes to electrostatic forces. An electron carries a negative charge, symbolized as \(-1.6 \times 10^{-19}\, C\). This value is extremely small, reflecting the minute charge each electron has. Despite the small charge, electrons play a critical role in enabling electrical phenomena.

Knowing the charge of an electron helps us to calculate various electrostatic effects, such as how many electrons contribute to a total charge. This quantity is essential for solving problems in electricity and magnetism, as it ties together concepts like electric fields and potential difference. So, anytime we have information about a net charge, we can determine the number of electrons involved simply by dividing the total charge by the charge of one electron.
Mass of an Electron
The mass of an electron, although tiny, is a crucial quantity in physics that affects the dynamics of particles. Each electron has a mass of approximately \(9.11 \times 10^{-31}\, kg\).

This small mass allows electrons to move rapidly and makes them well-suited to form the basis of electric currents. In electromagnetism problems, while we don't directly feel the impact of an electron's mass, it contributes to our understanding of phenomena like electricity conduction and particle acceleration.
  • The electron mass is paramount in determining how many excess electrons can accumulate without significantly affecting the overall mass of an object, such as a human body in our example.
  • Nonetheless, by accurately calculating the increase in mass due to electrons, we gain insights into the delicate balance of mass and charge in atomic-scale processes.
Excess Electrons
Excess electrons refer to additional negatively charged electrons that create a net negative charge in an object. When someone moves across a wool rug, for example, friction generates static electricity due to electrons being transferred from one material to another.

The number of excess electrons an object gains can be calculated by dividing the net charge by the charge of one electron. In our specific scenario, a person acquired a charge of \(-46 \mu C\), resulting in approximately \(2.875 \times 10^{14}\) excess electrons.
  • This method paints a clear picture of how static charges build up in real-world situations.
  • Tracking these excess electrons helps explain electrostatic shocks and other electrical interactions.
Thus, understanding the concept of excess electrons can deepen one's grasp of both theoretical and practical aspects of electrostatics.

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Most popular questions from this chapter

(II) A very thin line of charge lies along the \(x\) axis from \(x = - \infty\) to \(x = + \infty .\) Another similar line of charge lies along the \(y\) axis from \(y = - \infty\) to \(y = + \infty .\) Both lines have a uniform charge per length \(\lambda .\) Determine the resulting electric field magnitude and direction (relative to the \(x\) axis) at a point \(( x , y )\) in the first quadrant of the \(x y\) plane.

(II) Two charged spheres are 8.45\(\mathrm { cm }\) apart. They are moved, and the force on each of them is found to have been tripled. How far apart are they now?

(II) An electron has an initial velocity \(\vec { \mathbf { v } } _ { 0 } = \left( 8.0 \times 10 ^ { 4 } \mathrm { m } / \mathrm { s } \right) \hat { \mathbf { j } }\) . It enters a region where \(\quad \vec { \mathbf { E } } = ( 2.0 \hat { \mathbf { i } } + 8.0 \hat { \mathbf { j } } ) \times 10 ^ { 4 } \mathrm { N } / \mathrm { C }\) (a) Determine the vector acceleration of the electron as a function of time. \(( b )\) At what angle \(\theta\) is it moving (relative to its initial direction) at \(t = 1.0 \mathrm { ns } ?\)

(I) What is the electric field at a point when the force on a \(1.25 - \mu \mathrm { C }\) charge placed at that point is \(\vec { \mathbf { F } } = ( 3.0 \hat { \mathbf { i } } - 3.9 \hat { \mathbf { j } } ) \times 10 ^ { - 3 } \mathbf { N } ?\)

Four equal positive point charges, each of charge \(8.0 \mu \mathrm { C } ,\) are at the corners of a square of side 9.2\(\mathrm { cm } .\) What charge should be placed at the center of the square so that all charges are at equilibrium? Is this a stable or unstable equilibrium in the plane?

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