Power and Resistance: Relation between 100W, 60W and 40W Bulb Resistances

[erisedk]

Homework Statement

Incandescent bulbs are designed by keeping in mind that the resistance of their filament increases with an increase in temperature. If at room temperature, 100 W, 60 W and 40 W bulbs have filament resistances R₁₀₀, R₆₀ and R₄₀, respectively, the relation between these resistances is:

(A) $\dfrac{1}{R_{100}} = \dfrac{1}{R_{40}} + \dfrac{1}{R_{60}}$

(B) $R_{100} = R_{40} + R_{60}$

(C) $R_{100} > R_{60} > R_{40}$

(D) $\dfrac{1}{R_{100}} > \dfrac{1}{R_{60}} > \dfrac{1}{R_{40}}$

Homework Equations

P = $I^2R$
P = $\frac{V^2}{R}$

The Attempt at a Solution

I am extremely confused on this one. I think if we connected them all in series, which would mean that the current flowing through all three bulbs is the same, the resistance of each of them would increase, so power would increase but then this would decrease the current, so the power would still be the same etc. Then I tried similar things with fixed voltages, but I keep going around in circles, so if anyone could give me a starting point?
Thank you.

 

[CWatters]

Forget connecting them in series. Just think about each bulb connected to the same voltage V on it's own.

What is the equation that relates the power P, resistance R and voltage V ?

 

[CWatters]

Actually I see you quoted the relevant equation

P = V²/R

Rearrange that to give three equations for R₁₀₀, R₆₀ and R₄₀.

 

[erisedk]

That would give option A

 

[erisedk]

I did that but don't you think the first line of the problem should be used somehow?

 

[erisedk]

The one about resistance is dependent on temperature?

 

[CWatters]

That would give option A

Answer A is a trap :-)

Have a think about the other three. Two are easy to disprove.

 

[erisedk]

Yeah, (B) and (C) are pretty obviously wrong. (D) is surely right, and it is indeed the answer. But why is A wrong?

 

[CWatters]

A is only correct if you ignore that bit about the resistance increasing with temperature. A bulb that burns 100W will be hotter than a 60 or 40W.

It looks like the examiner put an almost correct answer first on the list to trap people that don't bother to check if other answers are even more correct. It's no coincidence the right answer is at the bottom of the list :-)

 

[erisedk]

Thanks! :)