Problem 151 - Master Resource Book in JEE Main Physics

Question

A conductor has been given a charge \(-3 \times 10^{-7} \mathrm{C}\) by transferring electron. Mass increase (in \(\mathrm{kg}\) ) of the conductor and the number of electrons added to the conductor are respectively, \(\quad\) [AMU Engg. 2010] (a) \(2 \times 10^{-16}\) and \(2 \times 10^{31}\) (b) \(5 \times 10^{-31}\) and \(5 \times 10^{19}\) (c) \(3 \times 10^{-19}\) and \(9 \times 10^{16}\) (d) \(2 \times 10^{-18}\) and \(2 \times 10^{12}\)

Step-by-Step Solution

Answer

The mass increase is \(2 \times 10^{-18} \mathrm{kg}\) and the number of electrons added is \(2 \times 10^{12}\).

Step 1: Calculate the Number of Electrons

An electron carries a charge of \( e = -1.6 \times 10^{-19} \mathrm{C} \). To find the number of electrons \( n \) transferred, we use the formula \( n = \frac{Q}{e} \), where \( Q = -3 \times 10^{-7} \mathrm{C} \). Thus, \( n = \frac{-3 \times 10^{-7}}{-1.6 \times 10^{-19}} \). Calculating this gives \( n = 1.875 \times 10^{12} \) electrons, which we can approximate to \( 2 \times 10^{12} \) electrons.

Step 2: Calculate Mass Increase

The mass of one electron is approximately \( 9.1 \times 10^{-31} \mathrm{kg} \). Using the number of electrons calculated in Step 1 \( (n = 2 \times 10^{12}) \), the total mass increase \( \Delta m \) can be calculated using \( \Delta m = n \times \text{mass of one electron} \). Thus, \( \Delta m = 2 \times 10^{12} \times 9.1 \times 10^{-31} = 1.82 \times 10^{-18} \mathrm{kg} \), which approximately equals \( 2 \times 10^{-18} \mathrm{kg} \).

Key Concepts

Charge Transfer

In electrostatics, charge transfer refers to the process of moving electrical charge from one object to another. This movement of charge can occur by various means, but in this context, we're looking at the transfer caused by adding or removing electrons. When a conductor receives a charge of

• The charge is given in coulombs (C), and represents an excess or deficit of electrons.
• In the given problem, the conductor is charged with
• negative charges, signifying an addition of electrons.

When electrons are transferred to a conductor: - The total charge - With a negative sign indicates extra electrons. Understanding the charge transfer is important because it sets the basis for calculating both the number of electrons transferred and subsequent changes, such as mass increase.

Electron Mass

The mass of an electron is a pivotal concept when addressing questions about mass changes due to charge transfer. Each electron has a small mass, approximately

• Electron mass: \( 9.1 \times 10^{-31} \mathrm{kg} \)

For this problem, understanding the electron mass allows you to determine how the mass of the conductor increases after receiving additional electrons. It's essential to comprehend that:- Even though the mass of each electron is minute, the cumulative effect of millions or billions of electrons can result in a noticeable change in mass.- Knowing the number of electrons involved (from the charge transfer) makes it straightforward to calculate the total mass increase. Thus, by multiplying the electron mass by the number of transferred electrons, you quantify the increase in mass precisely.

Calculation of Electrons

A crucial aspect of this exercise is determining how many electrons are involved in the charge transfer. Given the total charge and knowing the charge of each electron, you can calculate how many electrons constitute the charge.To solve for the number of electrons - Use the formula: \[ n = \frac{Q}{e}\]where:

• \( n \) is the number of electrons,
• \( Q = -3 \times 10^{-7} \mathrm{C} \) is the total charge, and
• \( e = -1.6 \times 10^{-19} \mathrm{C} \) is the charge per electron.

By performing this calculation:- It results in finding approximately \( \approx 2 \times 10^{12} \) electrons,- This number accounts for the excess charge of electrons that were added to the conductor.Understanding these calculations is vital for comprehending not just the number of particles involved, but also the electrostatic principles governing their behavior.