Three resistors, each with a resistance 𝑅, are connected in different ways as shown in the diagrams one through four. Which figure gives the smallest value of resistance between 𝑋 and 𝑌?
In figures one, two, three, and four, we see these resistors arranged between points marked 𝑋 and 𝑌 in each diagram. To solve for the figure which gives the smallest value of resistance between these two points, we can redraw each one. Figure one can be redrawn to show 𝑋 then a single resistor followed by two resistors arranged in parallel then point 𝑌.
Figure two can be redrawn as 𝑋 and 𝑌 separated by two parallel branches one with resistance 𝑅 and the other with resistance two 𝑅. In figure three, 𝑋 and 𝑌 are separated by a resistance of 𝑅. And there is a resistance of two 𝑅 that leads to a break in the circuit. And finally, in figure four, we can combine all three of the resistors in series so that 𝑋 and 𝑌 are separated by three 𝑅 of resistance.
Now that we’ve drawn simplified circuits for each diagram, we can calculate the resistance between points 𝑋 and 𝑌 in each one. In diagram one, 𝑅 𝑋𝑌 is equal to 𝑅, the value of the resistor in series, plus the value of the combined parallel resistors which is 𝑅 squared, their product, divided by two 𝑅, their sum. In this fraction, one factor of 𝑅 cancels out so that 𝑅 sub 𝑋𝑌 is equal to 𝑅 plus 𝑅 over two or three-halves 𝑅. That’s for diagram one.
For diagram two, 𝑅 sub 𝑋𝑌 is equal to the product of the two resistances in parallel: two 𝑅 times 𝑅 all divided by their sum which is three 𝑅. Once again, we see a factor of 𝑅 cancel from numerator and denominator. In total then, 𝑅 sub 𝑋𝑌 for diagram two is two-thirds 𝑅.
Moving on to diagram three, in this case, 𝑋 and 𝑌 are separated by one resistor of resistance value 𝑅. We write down that value then for 𝑅 sub 𝑋𝑌 for this diagram and move on to our final diagram number four. In this final diagram, 𝑅 sub 𝑋𝑌 is equal to three 𝑅.
Comparing these four values, we find that 𝑅 sub 𝑋𝑌 equalling two-thirds 𝑅 is the smallest of the bunch. That means the figure with the smallest resistance between points 𝑋 and 𝑌 is figure two.