Final answer:
Replacing the bulb filament with one that is twice as long and twice the diameter will cause an increase in resistance, resulting in lower current and decreased brightness because electric power is proportional to the current.
Step-by-step explanation:
When the filament of a bulb, which is connected to the same battery, is replaced with a filament of the same material but is twice as long and twice the diameter, the resistance of the filament will increase, affecting the brightness of the bulb.
Resistance (R) is given by ρL/A, where ρ is the resistivity of the material, L is the length of the conductor, and A is the cross-sectional area. The new filament has four times the original cross-sectional area, due to the doubling of the diameter.
However, the new filament's resistance will increase by a factor of eight because the resistance is directly proportional to the length (doubling the length doubles the resistance), and inversely proportional to the cross-sectional area (doubling the diameter, which quadruples the cross-sectional area, reduces the resistance by a factor of four).
Therefore, the net change in resistance is an increase by a factor of two (eight divided by four).
According to Ohm's law (V = IR), if the resistance increases and the voltage remains constant, the current through the circuit will decrease. Consequently, because electric power (P) is calculated as P = VI or P = I²R, a decrease in current results in a decrease in power, which in turn, reduces the bulb's brightness. So, with a higher resistance and lower current, the bulb will shine less brightly when connected to the same voltage.