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Chapter 25: Problem 13

What would happen to the drift velocity of electrons in a wire if the resistance due to collisions between the electrons and the atoms in the crystal lattice of the metal disappeared?

Short Answer

Expert verified
Answer: When the resistance due to collisions between electrons and atoms in the crystal lattice is removed, the drift velocity of the electrons in the wire would increase as a result of fewer collisions and more efficient movement of electrons through the wire.

Step by step solution

01

Understand the drift velocity of electrons in a wire

Drift velocity is the average velocity of electrons in a conductive material like a wire, due to an applied electric field. This drift velocity is quite slow as the electrons are constantly colliding with the atoms in the metal, which cause resistance.
02

Relationship between drift velocity and resistance

The drift velocity (v_d) of electrons in a wire can be calculated using the formula: v_d = I / (n * e * A) where I is the current through the wire, n is the number of electrons per unit volume, e is the charge of an electron, and A is the cross-sectional area of the wire. The resistance (R) of the wire is calculated using Ohm's law: R = V / I where V is the voltage across the wire and I is the current through the wire. In a metal, the resistance is mainly due to collisions between the electrons and the atoms in the lattice. These collisions slow down the electrons and decrease their overall drift velocity.
03

Effect of removing resistance due to collisions

If the resistance due to collisions between electrons and atoms in the crystal lattice were to disappear, the electrons would no longer be slowed down in their path. This means that they would experience fewer collisions and would be able to move more freely through the metal. As a result, the drift velocity of electrons in the wire would increase. In other words, without the resistance due to collisions, the electrons would move more efficiently through the wire, causing a decrease in the time it takes for them to travel through the conductive material. Consequently, the drift velocity of the electrons would increase.
04

Conclusion

In conclusion, if the resistance due to collisions between electrons and atoms in the crystal lattice of a metal were to disappear, the drift velocity of the electrons would increase as a result of fewer collisions and more efficient movement of electrons through the wire.

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Most popular questions from this chapter

A potential difference of \(V=0.500 \mathrm{~V}\) is applied across a block of silicon with resistivity \(8.70 \cdot 10^{-4} \Omega \mathrm{m}\). As indicated in the figure, the dimensions of the silicon block are width \(a=2.00 \mathrm{~mm}\) and length \(L=15.0 \mathrm{~cm} .\) The resistance of the silicon block is \(50.0 \Omega\), and the density of charge carriers is \(1.23 \cdot 10^{23} \mathrm{~m}^{-3}\) Assume that the current density in the block is uniform and that current flows in silicon according to Ohm's Law. The total length of 0.500 -mm-diameter copper wire in the circuit is \(75.0 \mathrm{~cm},\) and the resistivity of copper is \(1.69 \cdot 10^{-8} \Omega \mathrm{m}\) a) What is the resistance, \(R_{w}\) of the copper wire? b) What are the direction and the magnitude of the electric current, \(i\), in the block? c) What is the thickness, \(b\), of the block? d) On average, how long does it take an electron to pass from one end of the block to the other? \(?\) e) How much power, \(P\), is dissipated by the block? f) In what form of energy does this dissipated power appear?

A charged-particle beam is used to inject a charge, \(Q_{0}\), into a small, irregularly shaped region (not a cavity, just some region within the solid block) in the interior of a block of ohmic material with conductivity \(\sigma\) and permittivity \(\epsilon\) at time \(t=0\). Eventually, all the injected charge will move to the outer surface of the block, but how quickly? a) Derive a differential equation for the charge, \(Q(t)\), in the injection region as a function of time. b) Solve the equation from part (a) to find \(Q(t)\) for all \(t \geq 0\). c) For copper, a good conductor, and for quartz (crystalline \(\mathrm{SiO}_{2}\) ), an insulator, calculate the time for the charge in the injection region to decrease by half. Look up the necessary values. Assume that the effective "dielectric constant" of copper is \(1.00000 .\)

A \(12.0 \mathrm{~V}\) battery with an internal resistance \(R_{\mathrm{j}}=4.00 \Omega\) is attached across an external resistor of resistance \(R\). Find the maximum power that can be delivered to the resistor.

Show that the drift speed of free electrons in a wire does not depend on the cross-sectional area of the wire.

A material is said to be ohmic if an electric field, \(\vec{E}\), in the material gives rise to current density \(\vec{J}=\sigma \vec{E},\) where the conductivity, \(\sigma\), is a constant independent of \(\vec{E}\) or \(\vec{J}\). (This is the precise form of Ohm's Law.) Suppose in some material an electric field, \(\vec{E}\), produces current density, \(\vec{J},\) not necessarily related by Ohm's Law; that is, the material may or may not be ohmic. a) Calculate the rate of energy dissipation (sometimes called ohmic heating or joule heating) per unit volume in this material, in terms of \(\vec{E}\) and \(\vec{J}\). b) Express the result of part (a) in terms of \(\vec{E}\) alone and \(\vec{J}\) alone, for \(\vec{E}\) and \(\vec{J}\) related via Ohm's Law, that is, in an ohmic material with conductivity \(\sigma\) or resistivity \(\rho .\)
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