In the circuit below, what wake up to the brightness of the two light bulbs as soon as the switch is closed, connecting the 3rd light bulb to the circuit? You deserve to assume that light bulbs are prefer resistors, and that a bulb"s brightness is proportional to the present through it.

You are watching: Does bulb a get brighter, dimmer, or stay the same brightness?


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(a.) Bulbs 1 and also 2 obtain brighter(b.) Bulbs 1 and also 2 get dimmer(c.) bulb 1 it s okay brighter and bulb 2 gets dimmer(d.) bulb 1 gets dimmer and bulb 2 gets brighter(e.) Bulbs 1 and also 2 stay the same
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This is a conceptual problem, us don"t have any type of numbers. We are told, however, the the bulbs act favor resistors, and that brightness is proportional to the current. This way we have the right to redraw our figure and also start investigating what will happen.
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We want to know about current, yet we are given voltage (the source powering the bulbs). So we"re not certain if VDR or CDR will assist us just yet. What has readjusted between these 2 states? We put in one more resitor, so over there was certainly a change in identical resistance. We added another facet in parallel with resistor 2. And also we understand that including an element in parallel lowers resistance.
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Since resistance was lowered, but voltage remained the same, that method current has to rise come compensate.the full current through the entirety resistor network is equal to the present through aspect 1, since facet 1 is in series with the remainder of the circuit.The existing in element 1 rises, due to the fact that the full current rose.Bulb 1 gets brighterThis eliminates answer B, D, and also E
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Now we have to number out what happens to pear 2 without understanding anything around the new resistance exept that it is positiveWe found out in the previous ar that facet 1 has much more current. This means, by Ohm"s law, that it is likewise consuming more voltage.If aspect 1 is consuming much more voltage, climate that means that the remainder of the circuit (element 2 and the brand-new element) must have a lower voltage drop than before, because the 2 must add up to the resource in either case.And resistor 2 is in a parallel configuration. The voltage across a parallel construction is the same to the voltage across each that the elements. The voltage simply dropped.And with reduced voltage comes reduced current, and with lower present comes reduced brightness.
Therefore, bulb 2 gets dimmer
For those unsatisfied through the conceptual analysis this is the full mathematical proof for the difficulty above.
<eginalign&R_open&=&R_1+R_2&\ \&I_open R_open&=&V_B&\\&I_open&=&fracV_BR_open&\\&I_open&=&fracV_BR_1+R_2&\\&I_1,open&=&fracV_BR_1+R_2&\\&I_2,open&=&fracV_BR_1+R_2&endalign>
First, let"s totally anayze the initial, open switch stateI"ll use (R_open) to show the total equivalent resistance as soon as the move is open. (R_1) and (R_2) room the resistors the the two bulbs (V_B) is the voltage of the resource (presumably a battery)(I_open) is the existing through the battery once the switch is open.(I_1,open )is the current through bulb 1 when the move is openSince (R_1) and (R_2) are in series, castle both have actually currents equal to the battery current.
<eginalign&R_closed&=&R_1+fracR_2 R_3R_2+R_3&\\&I_closed R_closed&=&V_B&\\&I_closed &=& fracV_BR_closed&\\&I_closed &=& fracV_BR_1+fracR_2 R_3R_2+R_3&\\&I_1,closed&=&I_closed=fracV_BR_1+fracR_2 R_3R_2+R_3&\\&I_2,closed&=&I_1,closed fracR_3R_3+R_2&\\& &=&fracV_BR_1+fracR_2 R_3R_2+R_3 fracR_3R_3+R_2&endalign>
Now we perform the same evaluation on the close up door switch. Despite it to be unlabeled in the original problem, I will label the brand-new resistor (R_3).The current coming out of the battery currently uses the brand-new resistance. (I_1) matches this, yet (I_2) does not, since there is now an extra route or the current. The existing Divider ascendancy gives us the break-up of (I_closed) in between (R_2) and (R_3)
Now the we have actually expressions, we have the right to compare. First we to compare (I_1,open) to (I_1,closed).If your psychological math is strong, friend can notification that the denominator in the expression because that (I_1,closed) is definately less than the one because that (I_1,open), and division by a smaller quantity gives a larger finish result.
<eginalign&I_1,closed&=&fracV_BR_1+fracR_2 R_3R_2+R_3&\&I_1,closed&=&fracV_BR_1+fracR_2 R_3R_2+R_3 fracR_1+R_2R_1+R_2&\&I_1,closed&=&fracV_BR_1+R_2 fracR_1+R_2R_1+fracR_2 R_3R_2+R_3&\&I_1,closed&=&I_1,openfracR_1+R_2R_1+fracR_2 R_3R_2+R_3&\&I_1,closed&=&I_1,openfracR_1+R_2R_1+fracR_2 R_3R_2+R_3 fracR_2+R_3R_2+R_3&\&I_1,closed&=&I_1,openfrac(R_1+R_2)(R_2+R_3)R_1(R_2+R_3)+R_2 R_3 &\&I_1,closed&=&I_1,openunderbracefrac(R_1 R_2+R_2 R_3 +R_3 R_1) +R_2^2(R_1 R_2+R_1 R_3+R_2 R_3)_>1&endalign>
If that dispute isn"t convincing, we can rewrite the expression because that I_1,closed to do it have actually a much more obvious relationship to (I_1,open)We can present (I_1,closed) to be same to (I_1,open) time some huge factor, and that aspect is absolutely bigger than 1. Two things on top, among them top top bottom. Pear 1 becomes brighter after the move is closed

(I_2,open=fracV_BR_1+R_2)Vs(I_2,closed=fracV_BR_1+fracR_2 R_3R_2+R_3 fracR_3R_3+R_2)
<eginalign&I_2,closed&=&fracV_BR_1+fracR_2 R_3R_2+R_3 fracR_3R_3+R_2&\&I_2,closed&=&fracV_BR_1+R_2 fracR_1+R_2R_1+fracR_2 R_3R_2+R_3 fracR_3R_3+R_2&\&I_2,closed&=&fracV_BR_1+R_2 fracR_1+R_2R_1 (R_2+R_3)+R_2 R_3 fracR_31&\&I_2,closed&=&I_2,openfracR_3 R_1+R_3 R_2R_1 R_2+R_1 R_3+R_2 R_3 &\&I_2,closed&=&I_2,open underbracefrac(R_3 R_1+R_3 R_2)R_1 R_2+(R_1 R_3+R_2 R_3) _{To execute this, we simply cheat. Us multiply through something same to 1, i m sorry gies united state the denominator we desire times a bunch of junkDistruibute the (R2+R3) termDistribute moreWe now have the original expression times a quantity that is certainly less than 1, because it is something separated by itself added to an additional positive number pear 2 gets dimmer.

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As can be seen, the qualitative way of solving things can obtain us a tiny result, like "increases" or "decreases" for fairly little effort. Come get precisely HOW much it increases, we need to do part math. And also by some math I mean a lot of tedious algebra.