The resistance of hot tungsten filament is about 10 times the cold resistance. What will be the resi

Q: Why does the resistance of the filament change with temperature?

Tungsten is a metal, and metals have a positive temperature coefficient of resistivity. As the temperature increases, the atoms in the metal lattice vibrate more vigorously. This causes the flowing electrons to collide more frequently with the lattice, thereby increasing the overall electrical resistance.

Q: Can I use the formula $P = I^2 R$ to solve this?

Yes, you can, but it requires an extra step. You would first need to find the current using $P = VI$ ($100 = 200 \times I$, so $I = 0.5\text{ A}$). Then, you could use $R = P / I^2$ ($100 / 0.5^2 = 400\,\Omega$). Using $P = V^2 / R$ is more direct since it uses the given values immediately.

Q: Why do the power and voltage ratings only apply to the hot state?

Manufacturers provide these ratings to tell you how the bulb will perform when it is actually doing its job (producing light). The bulb is designed to draw 100 W of power *only* when connected to a 200 V source and after it has reached its stable, high operating temperature.

Q: What would happen if we connected this bulb to a 100 V source instead of 200 V?

If connected to a lower voltage, the bulb wouldn't get as hot. Because it wouldn't reach its full operating temperature, its resistance would be lower than $400\,\Omega$ (but still higher than $40\,\Omega$). Consequently, it would draw less power and glow much dimmer.

Q: Is the resistance always exactly 10 times higher when hot?

Not exactly. The factor of 10 is an approximation used for this specific problem. In reality, the ratio depends on the exact material of the filament and its specific operating temperature. However, for standard incandescent tungsten bulbs, the hot resistance is typically 10 to 15 times greater than the cold resistance.

Question

The resistance of hot tungsten filament is about 10 times the cold resistance. What will be the resistance of 100 W and 200 V lamp, when not in use?

Options:

(A) $40 \,\Omega$

(B) $20 \,\Omega$

(C) $400 \,\Omega$

(D) $200 \,\Omega$

Sigma Percentile · Accepted Answer

Imagine you are holding a standard incandescent light bulb. When it sits in your hand, completely switched off, its tungsten filament is at room temperature. We refer to this state as its **cold state**.

However, the moment you screw it into a socket and flip the switch, electricity surges through it. The filament heats up to thousands of degrees, glowing brilliantly white. This is its **hot state**.

Because tungsten is a metal, its electrical resistance increases significantly as its temperature rises. The problem tells us a fascinating fact: the resistance of this filament when it is hot is roughly **10 times** its resistance when it is cold!

### Analyzing the Setup

When you look at the packaging of a light bulb, you will see ratings like **100 W** and **200 V**. It is crucial to understand that these ratings describe the bulb *only* when it is operating normally.

Therefore, the power $P = 100	ext{ W}$ and the voltage $V = 200	ext{ V}$ correspond to the bulb in its hot, glowing state. We can use these values to find the resistance of the hot filament, which we will call $R_{	ext{hot}}$.

### The Master Equation

To connect power, voltage, and resistance, we use the fundamental electrical power formula:

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

Since we are dealing with the operating state, we can rewrite this specifically for our hot filament:

$$R_{	ext{hot}} = \frac{V^2}{P}$$

### Final Calculation

Let's substitute the given values into our rearranged equation:

$$R_{	ext{hot}} = \frac{200 	imes 200}{100}$$

$$R_{	ext{hot}} = \frac{40000}{100} = 400 \,\Omega$$

So, when the bulb is shining brightly, its resistance is $400 \,\Omega$. But the question asks for the resistance when the lamp is **not in use**—which is the cold resistance, $R_{	ext{cold}}$.

We were given the relationship:

$$R_{	ext{hot}} = 10 	imes R_{	ext{cold}}$$

To find the cold resistance, we simply divide the hot resistance by 10:

$$R_{	ext{cold}} = \frac{R_{	ext{hot}}}{10}$$

$$R_{	ext{cold}} = \frac{400}{10} = 40 \,\Omega$$

The resistance of the lamp when it is switched off is $40 \,\Omega$. This highlights a beautiful physical reality: the properties of materials can change drastically depending on their thermal state!

Animated Solution for Physics - Current Electricity: The resistance of hot tungsten filament is about 10 times the cold resistance. What will be the resistance of 100 W and 200 V lamp, when not in use?

Select Answer:

Visualized Solution

Understanding the States of the Filament

Power Rating Formula

Calculating Hot Resistance

Computing Hot Resistance

Calculating Cold Resistance

Final Conclusion

The Sigma Insight: Ohm's Law, Resistance and Electrical Power

Analyzing the Setup

The Master Equation

Final Calculation

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