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Chapter 12: Problem 1742

The drift velocity of free electrons in a conductor is \(\mathrm{V}\), when a current. \(I\) is flowing in it If both the radius and current are doubled, then drift velocity will be. (A) \((\mathrm{V} / 4)\) (B) \((\mathrm{V} / 2)\) (C) \(4 \mathrm{~V}\) (D) \(2 \mathrm{~V}\)

Short Answer

Expert verified
The new drift velocity after doubling both the radius and current of the conductor is \((\mathrm{V} / 4)\).

Step by step solution

01

Write down the formula relating drift velocity, current, and conductor properties

The formula we need is given by: \[I = nAvq\] where: - \(I\) is the electric current, - \(n\) is the number of free electrons per unit volume, - \(A\) is the cross-sectional area of the conductor, - \(v\) is the drift velocity of the free electrons, and - \(q\) is the charge of an electron.
02

Analyze the changes in the parameters

Given that both the radius and the current are doubled, we have the following changes: 1. The new radius is \(2r\), where \(r\) is the initial radius. 2. The new current is \(2I\), where \(I\) is the initial current. We need to find out how these changes affect the drift velocity.
03

Calculate the new cross-sectional area

The cross-sectional area of the conductor is given by the formula: \[A = \pi r^2\] When the radius is doubled (to \(2r\)), the new cross-sectional area becomes: \[A' = \pi (2r)^2 = 4\pi r^2\] Now, the new area is four times the initial area: \(A' = 4A\).
04

Write the formula for the new drift velocity

We can write the formula for the new drift velocity (\(v'\)) in terms of the new current and area: \[2I = n(4A)v'q\] Now, we need to solve this equation for \(v'\).
05

Calculate the new drift velocity

First, rearrange the formula to solve for \(v'\): \[v' = \frac{1}{4} \cdot \frac{2I}{nAq}\] Since \(I = nAvq\), we have: \[v' = \frac{1}{4} \cdot \frac{nAvq}{nAq} = \frac{1}{4}v\]
06

Choose the correct option

According to our calculation, the new drift velocity is: \[v' = \frac{1}{4}v\] Therefore, the correct answer is (A) \((\mathrm{V} / 4)\).

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Most popular questions from this chapter

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