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Chapter 25: Problem 23

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

Short Answer

Expert verified
Answer: No, the drift speed does not depend on the cross-sectional area. It is a function of the length of the wire (L) and the time (t) rather than its cross-sectional area (A).

Step by step solution

01

Recall the formula for drift speed

The drift speed (v_d) of free electrons in a wire can be expressed as: v_d = I / (n * A * q), where I is the current flowing through the wire, n is the number density of the free electrons (number of electrons per unit volume), A is the cross-sectional area of the wire, and q is the charge of an electron. Our goal is to show that the drift speed does not depend on the cross-sectional area (A) of the wire.
02

Express current in terms of charge and number of electrons

We know that the current flowing through a wire (I) can be expressed as the charge (Q) passing through the wire per unit time (t): I = Q / t, where Q = n * V * q. Here, V represents the volume of the wire through which the charge passes.
03

Relate the volume of the wire with its cross-sectional area

We can express the volume of the wire (V) as a product of its cross-sectional area (A) and length (L): V = A * L.
04

Substitute Q and V in the current formula

Now we substitute Q = n * V * q and V = A * L in the current formula. I = (n * A * L * q) / t.
05

Simplify the formula for drift speed

By rearranging the equation, we can find the drift speed (v_d) of the free electrons in the wire. v_d = I / (n * A * q) = (n * A * L * q) / (t * n * A * q) = L / t.
06

Conclude that drift speed does not depend on the cross-sectional area

From the above formula, we see that drift speed (v_d) is given by the ratio of length (L) to time (t), and there is no mention of the cross-sectional area (A) in this expression. Thus, we can conclude that the drift speed of free electrons in a wire does not depend on its cross-sectional area.

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Most popular questions from this chapter

Two identical light bulbs are connected to a battery. Will the light bulbs be brighter if they are connected in series or in parallel?

You are given two identical batteries and two pieces of wire. The red wire has a higher resistance than the black wire. You place the red wire across the terminals of one battery and the black wire across the terminals of the other battery. Which wire gets hotter?

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?

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.

The most common material used for sandpaper, silicon carbide, is also widely used in electrical applications. One common device is a tubular resistor made of a special grade of silicon carbide called carborundum. A particular carborundum resistor (see the figure) consists of a thick-walled cylindrical shell (a pipe) of inner radius \(a=\) \(1.50 \mathrm{~cm},\) outer radius \(b=2.50 \mathrm{~cm},\) and length \(L=60.0 \mathrm{~cm} .\) The resistance of this carborundum resistor at \(20 .{ }^{\circ} \mathrm{C}\) is \(1.00 \Omega\). a) Calculate the resistivity of carborundum at room temperature. Compare this to the resistivities of the most commonly used conductors (copper, aluminum, and silver). b) Carborundum has a high temperature coefficient of resistivity: \(\alpha=2.14 \cdot 10^{-3} \mathrm{~K}^{-1} .\) If, in a particular application, the carborundum resistor heats up to \(300 .{ }^{\circ} \mathrm{C},\) what is the percentage change in its resistance between room temperature \(\left(20 .{ }^{\circ} \mathrm{C}\right)\) and this operating temperature?
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