Animated Solution for Physics - Current Electricity: The resistance of a bulb filament is $100\Omega$ at a temperature of $100^{\circ }\text{C}$. If its temperature coefficient of resistance is $0.005{\text{ per }}^{\circ }\text{C}$, its resistance will become $200\Omega$ at a temperature of

Imagine you are looking at a classic incandescent light bulb. When you flip the switch, electricity surges through the thin tungsten filament. It doesn't just light up instantly; it heats up tremendously. As the temperature of this metal filament rises, the atoms vibrate more vigorously, making it harder for electrons to flow. In physics terms, the electrical resistance of the metal increases with temperature.

This problem asks us to find the exact temperature at which the resistance of a specific bulb filament doubles from $100\Omega$ to $200\Omega$.

For most metallic conductors over a reasonable range of temperatures, the relationship between resistance and temperature is beautifully linear. We describe this using the master equation:

Here, $R_T$ is the resistance at a temperature $T$ (in ${}^{\circ }\text{C}$), $R_0$ is the base resistance at $0^{\circ }\text{C}$, and $\alpha$ is the temperature coefficient of resistance.

In our problem, we are given $\alpha =0.005{}^{\circ }{\text{C}}^{-1}$. However, we are not given $R_0$. This might seem like a roadblock, but let's set up our conditions and see what happens.

We are given two distinct states for the bulb filament.

Condition 1: At a temperature $T_1=100^{\circ }\text{C}$, the resistance is $R_1=100\Omega$. Plugging this into our master equation gives:

Condition 2: We want to find the unknown temperature, let's call it $x$, where the resistance becomes $R_2=200\Omega$. This gives us our second equation:

We now have a system of two equations with two variables: $R_0$ and $x$. Since $R_0$ is just an intermediate constant that we don't actually need to find, the most elegant mathematical move is to eliminate it entirely. How? By dividing the second equation by the first!

Notice how the $R_0$ terms perfectly cancel each other out. This is a classic physics problem-solving technique: when you have an unknown scaling factor in two proportional states, division is your best friend.

Let's simplify the resulting equation. The left side simplifies to $2$. On the right side, the denominator becomes $1+0.5=1.5$.

Now, we multiply both sides by $1.5$ to clear the fraction:

Subtracting $1$ from both sides leaves us with:

Finally, to isolate $x$, we divide $2$ by $0.005$. A quick mental math trick is to multiply the numerator and denominator by $1000$:

Thus, the filament must reach a scorching $400^{\circ }\text{C}$ for its resistance to double to $200\Omega$.