Question

A certain computer chip that is about the size of a postage stamp contains about a million transistors. If the transistors are square, what must be their maximum dimension? (Note: Devices other than transistors are also on the chip, and there must be a room for the interconnections among the circuit elements. Transistors smaller than are now commonly and inexpensively fabricated.)

Answer 1

Hint
The transistors are square, and will therefore cover an area equal to the square of their sides. The total area of the computer chip must be equal to the area occupied by a single transistor multiplied by the total number of transistors.
 $A=n{l}^2$
Where A is the area of the computer chip
n is the number of transistors in the chip
l is the side length of a single chip

Complete step by step answer
It is given in the question that,
Size of the computer chip $=2.54cm\times 2.22cm$
Therefore the area A is given by,
 $A=2.54\times 2.22c{m}^2$
 $A=5.6388c{m}^2$
Writing this in metres we have,
 $A=5.6388\times {10}^{-4}{m}^2$
Now if the total number of transistors is $3.5million$
 $n=3.5\times {10}^6$
Therefore length is given by,
 $l=\sqrt{{\displaystyle \frac{A}{n}}}$
 $l=\sqrt{{\displaystyle \frac{5.6388\times {10}^{-4}}{3.5\times {10}^6}}}$
 $l=\sqrt{1.611\times {10}^{-10}}$
 $l=1.269\times {10}^{-5}m$
The length of the square shaped transistor is 12 $\mu m$ .

Additional Information
Transistors are widely used in electronic devices because they can act as a switch. In a computer program there are conditions and each condition has a different outcome. A transistor can act as a switch between these outcomes, thus is an integral part of electronic devices including computers. Many other components like diodes, zener diodes, resistors and capacitors are also used to create the chip of a computer. These are arranged in patterns called “gates”, each gate has an individual function for decision making. For example, an AND gate is true only when both of the inputs are 1, and so on. Apart from AND gate there are also many other gates like OR,NOR, XOR, NAND, and so on.

Note
The length of transistor should be lower than $12\mu m$ because 12 is the maximum size that can be accommodated into a computer chip, if the other components are also to be included, the size of the chips will go down.