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# You are given a number of$10 \Omega$resistors, each capable of dissipating only 1.0 W without being destroyed. What is the minimum number of such resistors that you need to combine in series or in parallel to make a$10 \Omega$resistance that is capable of dissipating at least 12 W?

Solution

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Given:

$r=10\,\,\rm{\Omega}$

$p=1\,\,\rm{W}$

Net resistance increases in series connection and decreases in parallel connection. So for the net resistance to be equal to the individual resistance there must be a combination of both series and parallel.

There are not many combinations that can fulfill that so we can go through each First case consists of 2 series of 2 resistors in parallel:

$R_a=\frac{10}{2}+\frac{10}{2}=10\,\,\rm{\Omega}$

$P_a=i^2R_a$

$P_a=i^2\cdot 10$

From the condition that $P>12\,\,\rm{W}$ we get:

$i^2>1.2$

Also, power dissipation through each resistor

$P_r=(\frac{i}{2})^2R$

$P_r=\frac{i^2}{4}R$

$P_r=\frac{1.2^2}{4}\cdot 10$

$P_r=3\,\,\rm{W}$

This condition is not met, therefore this layout will not work.

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