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Chapter 18: Problem 17

A strip of doped silicon \(260 \mu \mathrm{m}\) wide contains $8.8 \times 10^{22}$ conduction electrons per cubic meter and an insignificant number of holes. When the strip carries a current of \(130 \mu \mathrm{A},\) the drift speed of the electrons is \(44 \mathrm{cm} / \mathrm{s} .\) What is the thickness of the strip?

Short Answer

Expert verified
Answer: The thickness of the silicon strip is approximately \(5.62 \mu \mathrm{m}\).

Step by step solution

01

Write down the given values

The given values are: Width of strip (W) = \(260 \mu \mathrm{m}\) Number of conduction electrons per unit volume (n) = \(8.8 \times 10^{22} \mathrm{m}^{-3}\) Current (I) = \(130 \mu \mathrm{A}\) Drift speed of electrons (v) = \(44 \mathrm{cm} / \mathrm{s}\) First, we need to convert the given values into base unit SI units.
02

Convert units

Convert the given values to SI units: Width (W) = \(260 \times 10^{-6} \mathrm{m}\) Current (I) = \(130 \times 10^{-6} \mathrm{A}\) Drift speed (v) = \(0.44 \mathrm{m} / \mathrm{s}\)
03

Consider the formula for current

The formula for the current in terms of electron drift speed and charge density is given by: $$I = nqAv$$ Where: I = current, n = number of charge carriers per unit volume, q = charge of an electron, A = cross-sectional area, and v = drift speed. We need to find the thickness (t) of the strip. The cross-sectional area of the strip is given by the product of the width (W) and thickness (t), so: $$A = W \cdot t$$ We will use this equation to express the thickness in terms of the other variables and the current.
04

Express thickness in terms of other variables and current

Now we will express the thickness (t) in terms of other variables and current. Substitute the cross-sectional area formula into the current formula: $$I = nq(W \cdot t)v$$ Now, isolate the thickness variable t: $$t = \frac{I}{nqWv}$$
05

Calculate the thickness of the silicon strip

Now we can use the values of the current, charge carriers density, width, drift speed, and charge per electron to calculate the thickness. Charge of an electron (q) = \(1.6 \times 10^{-19} \mathrm{C}\) $$t = \frac{130 \times 10^{-6}}{(8.8 \times 10^{22})(1.6 \times 10^{-19})(260 \times 10^{-6})(0.44)}$$ Calculate the thickness: $$t \approx 5.62 \times 10^{-6} \mathrm{m}$$
06

Convert the thickness back into micrometers

Now, convert the thickness back into micrometers: $$t \approx 5.62 \times 10^{-6} \mathrm{m} \times \frac{10^6 \mu \mathrm{m}}{1 \mathrm{m}}$$ $$t \approx 5.62 \mu \mathrm{m}$$ The thickness of the silicon strip is approximately \(5.62 \mu \mathrm{m}\).

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Most popular questions from this chapter

A 1.5 -horsepower motor operates on \(120 \mathrm{V}\). Ignoring \(l^{2} R\) losses, how much current does it draw?
A 20 - \(\mu\) F capacitor is discharged through a 5 -k\Omega resistor. The initial charge on the capacitor is \(200 \mu \mathrm{C}\) (a) Sketch a graph of the current through the resistor as a function of time. Label both axes with numbers and units. (b) What is the initial power dissipated in the resistor? (c) What is the total energy dissipated?
Capacitors are used in many applications where one needs to supply a short burst of relatively large current. A 100.0 - \(\mu\) F capacitor in an electronic flash lamp supplies a burst of current that dissipates \(20.0 \mathrm{J}\) of energy (as light and heat) in the lamp. (a) To what potential difference must the capacitor initially be charged? (b) What is its initial charge? (c) Approximately what is the resistance of the lamp if the current reaches \(5.0 \%\) of its original value in $2.0 \mathrm{ms} ?$
What is the energy stored in a small battery if it can move \(675 \mathrm{C}\) through a potential difference of \(1.20 \mathrm{V} ?\)
(a) In a charging \(R C\) circuit, how many time constants have elapsed when the capacitor has \(99.0 \%\) of its final charge? (b) How many time constants have elapsed when the capacitor has \(99.90 \%\) of its final charge? (c) How many time constants have elapsed when the current has \(1.0 \%\) of its initial value?
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