Question

You have six 1.5 kω resistors.


part a


how can you connect them to produce a total equivalent resistance of 1.0 kω ?

Answer 1

Answer:

Have three pairs of 1.5kΩ connected in series and then connect all three in parallel.

Explanation:

This problem is kind of hard to explain and is based off my experience and intuition. I'll show how I got the answer and then explain afterwards.

If you have three pairs of 1.5kΩ in series, you get 3 3.0kΩ resistor equivalent (resistors in series). Putting them in parallel, you use get a resistance of 1 kΩ as shown below:

$R_{eq} = (\frac{1}{R_{1}} + \frac{1}{R_{2}} +...+\frac{1}{R_{n}} )^{-1} \\R_{eq} = (\frac{1}{R_{1}} + \frac{1}{R_{2}} + \frac{1}{R_{3}} )^{-1}\\R_{eq} = (\frac{1}{3} + \frac{1}{3} + \frac{1}{3} )^{-1}\\ R_{eq}  = 1$

How I got here takes some math sense. Since 1.5kΩ is greater than 1.0kΩ, you know there is a parallel combination. Next I considered what are the possible combinations. Since two 1.5kΩ resistors yielded 3.0kΩ, and you have 3 pairs of 3.0kΩ resistor, the sum of parallel would be 1.0kΩ

Answer 2

We can connect six 1.5 kΩ resistors to produce a total equivalent resistance of 1 kΩ if we have two sets of 3 parallel resistors connected in series.

Since we have six 1.5 kΩ resistors and we wish to connect them to produe an equivalent resistance of 1 kΩ.

Since 1 kΩ = 0.5 kΩ + 0.5 kΩ, we need to produce two sets of resistors that would give us an equivalent resistance of 0.5 kΩ and we need to find the number of reistors in each set.

Equivalent resistance

Since we have 1.5 kΩ resistors and we require to produce an equivalent resistance 0.5 kΩ which is less than the individual resistance, we connect them in parallel. So, we find the number of resistors, n required.

For parallel resistors, the equivalent resistance is

1/R' = 1/R₁ + 1/R₂ + ... + 1/Rₙ

Since R = R₁ = R₂ = ... = Rₙ

1/R' = n/R

Number of resistors

So, n = R/R'

Since R = 1.5 kΩ and R' = 0.5 kΩ

n = R/R'

n = 1.5 kΩ/0.5 kΩ

n = 3

So, we have two sets of 3 parallel resistors to give an equivalent resistance of 0.5 kΩ each. We then connect these two sets in series to give an equivalent resistance of 1 kΩ since the equivalent resistance in series is the sum of their resistances.

So, we can connect six 1.5 kΩ resistors to produce a total equivalent resistance of 1 kΩ if we have two sets of 3 parallel resistors connected in series.

Learn more about equivalent resistance here:

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