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Chapter 12: Problem 1712

The drift velocity of free electrons through a conducting wire of radius \(\mathrm{r}\), carrying current \(\mathrm{I}\), is \(\mathrm{V}_{\mathrm{d}}\) if the same current is passed through a conductor of radius \(2 \mathrm{r}\) what will be the drift velocity? (A) \(\left(\mathrm{V}_{\mathrm{d}} / 4\right)\) (B) \(\mathrm{V}_{\mathrm{d}}\) (C) \(2 \mathrm{~V}_{\mathrm{d}}\) (D) \(24 \mathrm{~V}_{\mathrm{d}}\)

Short Answer

Expert verified
The drift velocity of the conductor with radius 2r is \(\frac{1}{4}V_{d}\), which corresponds to answer choice (A).

Step by step solution

01

Find the relationship between drift velocity, radius, and current

To find the relationship, we must first make use of the formula for current (I) in a wire, which is given by: \[I = nAvq\] where, - I: Current in the wire - n: the number density of electrons - A: cross-sectional area of the wire - v: drift velocity of the electrons - q: charge of an electron The cross-sectional area of a conducting wire is given by: \[A = \pi r^2\] Now, we can rewrite the current formula as follows: \[I = n(\pi r^2)vq\]
02

Find the drift velocity of the wire with radius r

We are given the drift velocity, Vd, for the wire with radius r. So, for this wire, the current formula can be written as: \[I = n(\pi r^2)V_{d}q\]
03

Find the drift velocity of the wire with radius 2r

We have to find the drift velocity, Vd', for the wire with radius 2r. Both wires carry the same current, I. So, we can write the current formula for the wire with radius 2r: \[I = n(\pi (2r)^2)V_{d'}q\]
04

Solve for the new drift velocity

Now we must simply solve for \(V_{d'}\), the drift velocity of the electrons in the wire with the radius 2r. Divide the current equation of wire 2r by the current equation of wire r: \[\frac{n(\pi (2r)^2)V_{d'}q}{n(\pi r^2)V_{d}q} = 1\] Notice that the number density n, electron charge q, and π will cancel each other out: \[\frac{(2r)^2V_{d'}}{r^2V_{d}} = 1\] Solve for \(V_{d'}\): \[V_{d'} = \frac{r^2V_{d}}{(2r)^2} = \frac{1}{4}V_{d}\]
05

Identify the correct answer

After solving for the new drift velocity, we have: \[V_{d'} = \frac{1}{4}V_{d}\] Therefore, the correct answer is (A) \(\left(\mathrm{V}_{\mathrm{d}} / 4\right)\).

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Most popular questions from this chapter

The potential difference between the terminals of a battery is $10 \mathrm{~V}\( and internal resistance \)1 \Omega\( drops to \)8 \mathrm{~V}$ when connected across an external resistor find the resistance of the external resistor. (A) \(40 \Omega\) (B) \(0.4 \Omega\) (C) \(4 \mathrm{M} \Omega\) (D) \(4 \Omega\)

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