You are assuming, incorrectly, that the current is the same in both cases. It's not.
Start by considering that the resistance of the incandescent bulb is a constant. You can calculate the value of the resistance from P_1 = V_1^2/R. Now that you have a value for R, you can use it to calculate a new value of power for a different voltage using P_2 = V_2^2/R. Alternatively you can divide the second equation by the first:
P_2/P_1 = (V_2^2)/(V_1^2)
From this you can see that if V_2 = (1/2) V_1, then P_2 = (1/4) P_1.
Another approach is to recognize from V=IR that if V is cut in half then I must be cut in half (assuming constant R). Then from P_2 = V_2 I_2 you have that both V_2 and I_2 are half their original values. Hence P_2 is 1/4 its original value.
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Last edited by ChipB; May 9th 2016 at 07:17 AM.
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