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Three identical resistors are connected in series. When a certain potential difference is applied across the combination, the total power dissipated is 27 $\mathrm{W}$ . What power would be dissipated if the three resistors were connected in parallel across the same potential difference?
$P=243 \mathrm{W}$
Physics 102 Electricity and Magnetism
Chapter 19
Current, Resistance, and Direct-Current Circuit
Electric Charge and Electric Field
Gauss's Law
Electric Potential
Capacitance and Dielectrics
Current, Resistance, and Electromotive Force
Direct-Current Circuits
Electromagnetic Induction
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Here it is given in the problem that 3 identical resistors are connected in series, so there is seated a total power of 27. What now we want to calculate the power dissipated by all these 3 resisters when they are conected in parallel with the same potential difference? So let this potential difference of now by using the relation power is equal to sucro voltage upon r equivalent, so we have r equivalent equal to square of v upon power. Here we have power equal to 27. What is in serious connection we have r equivalent is equal to r 1 plus r, 2 plus r 3 here r, 1 r, 2 and r 3 are having some resistance, so we have r equivalent equal to r plus r plus r. Let r is the resistance of a rester, so we have r equivalent is equal to 3 times r. Now we have 2 values of r equivalent 1 is equal to 3 r, and the other 1 is equal to v square upon 27. What now, by equating these 2 values, we have 3 r, is equal to v square upon 27. What so we have resistance equal to v square upon 81, so this is the resistance of each resistor. Now we want to connect the resisters in parallel. So, in parallel connection, we have reciprocal of equivalent resistance is equal to 1 over r 1. Here r 1 is equal to r plus 1 over r plus 1 over r. So we have 1 of our r equivalent is equal to 3 upon r. So we have r equivalent equal to r upon 3 now by putting the value of r here. So we have r equal is equal to here. R is here: r is v square upon 81 word upon 3, so we have r equivalent, in parallel connection, is equal to v square upon 243. Now the total power consumed by the 3 resisters in parallel connection is p equal to v square upon r equivalent, which is equal to v square upon r equivalent here r equivalent is v square upon 243 at which is equal to 243 word upon v scare multiply. V, v sc, so here we sugar and scar will be canceled, so we have power equal to 243 at so when these resistors are connected in parallel and they will dissipate a total power of 243 wot.
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