Some light bulbs have three power settings (not including...

Name: Problem 7, Chapter 10: University Physics
Uploaded: 2020-06-29T04:32:18-08:00
Duration: 3 min 45 s
Channel: Ren Jie Tuieng
Description: Some light bulbs have three power settings (not including zero), obtained from multiple filaments that are individually switched and wired in parallel. What is the minimum number of filaments needed for three power settings?

Some light bulbs have three power settings (not including zero), obtained from multiple filaments that are individually switched and wired in parallel. What is the minimum number of filaments needed for three power settings?

Key Concepts

Binary Combination Principle

This concept refers to the fact that if you have n independent binary (on/off) elements, you can construct up to 2^n different combinations of states. This mathematical property is fundamental in digital systems and combinatorics for efficiently encoding multiple configurations using the fewest components possible.

Exclusion of the Zero State

Often, the configuration where all elements are off is not included as an active state. Therefore, when determining the number of useful configurations, you subtract this state, resulting in 2^n - 1 effective states. Recognizing when and how to apply this exclusion is essential for optimizing the number of components needed to meet specific nonzero configuration requirements.

Transcript

00:01 Now this question we are given a light bulb and we want the light bulb to have three different power settings.

00:05 So the power of a light bulb is actually determined by the current that's flowing through it and of course its own resistance.

00:17 But we can control only the current and not the resistance of the light bulb.

00:23 And the way that we can control the power through the current would be to adjust the overall resistance of the entire circuit and the filaments act as resistors so having filaments would act as resistors that will adjust the current in our circuit and hence adjust the power setting for our light bulb so if you want three different power settings we will need a minimum number of two filaments now let me explain why and two filaments must be in parallel so they act as resistors so the circuit will be as such right and we'll have switches over here and over here let me explain why so for the first power setting let me say let's just say high so we want to have a high power so high current and that will be when both switches are closed and current can flow through.

01:39 In this case, the resistor, let us call it r and r prime, the total resistance from this circuit would be r1 over r plus 1 over r prime, inverse, which would be the smallest possible amount of resistors, sorry, resistance that is across these two points.

01:59 These two points, we'll have the smallest resistance when both switches are closed.

02:08 The second setting, medium, would probably be when one of the switches are open and the current can only pass through one.

02:21 So the resistors would be just one, r...

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