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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

A battery has a potential difference of \(14.50 \mathrm{~V}\) when it is not connected in a circuit. When a \(17.91-\Omega\) resistor is connected across the battery, the potential difference of the battery drops to \(12.68 \mathrm{~V}\). What is the internal resistance of the battery?

Two cylindrical wires, 1 and \(2,\) made of the same material, have the same resistance. If the length of wire 2 is twice that of wire 1 , what is the ratio of their cross-sectional areas, \(A_{1}\) and \(A_{2} ?\) a) \(A_{1} / A_{2}=2\) c) \(\mathrm{A}_{1} / \mathrm{A}_{2}=0.5\) b) \(A_{1} / A_{2}=4\) d) \(A_{1} / A_{2}=0.25\)

How much money will a homeowner owe an electric company if he turns on a 100.00 -W incandescent light bulb and leaves it on for an entire year? (Assume that the cost of electricity is \(\$ 0.12 / \mathrm{kW} \mathrm{h}\) and that the light bulb lasts that long.) The same amount of light can be provided by a 26.000-W compact fluorescent light bulb. What would it cost the homeowner to leave one of those on for a year?

A 34 -gauge copper wire, with a constant potential difference of \(0.10 \mathrm{~V}\) applied across its \(1.0 \mathrm{~m}\) length at room temperature \(\left(20 .{ }^{\circ} \mathrm{C}\right),\) is cooled to liquid nitrogen temperature \(\left(77 \mathrm{~K}=-196^{\circ} \mathrm{C}\right)\) a) Determine the percentage change in the wire's resistance during the drop in temperature. b) Determine the percentage change in current flowing in the wire. c) Compare the drift speeds of the electrons at the two temperatures.

What is the current density in an aluminum wire having a radius of \(1.00 \mathrm{~mm}\) and carrying a current of \(1.00 \mathrm{~mA}\) ? What is the drift speed of the electrons carrying this current? The density of aluminum is \(2.70 \cdot 10^{3} \mathrm{~kg} / \mathrm{m}^{3},\) and 1 mole of aluminum has a mass of \(26.98 \mathrm{~g}\). There is one conduction electron per atom in aluminum.
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