Incandescent bulbs are designed by keeping in mind that the resistance of their filament increases with the increase in temperature. If at room temperature, `100W, 60W and 40W` bulbs have filament resistances `R_(100), R_(60) and R_(40)`, respectively, the relation between these resistances is

Options

A. `1/R_(100) = 1/R_(40) + 1/R_(60)`
B. `R_(100) = R_(40) + R_(60)`
C. `R_(100)gtR_(60)gtR_(40)`
D. `1/R_(100) gt 1/R_(60) gt 1/R_(40)`

Explanation

To solve the problem, we need to understand the relationship between power, voltage, and resistance in incandescent bulbs. The power \( P \) of a bulb can be expressed using Ohm's law and the formula for electrical power:

\[ P = \frac{V^2}{R} \]

Where:
- \( P \) is the power of the bulb,
- \( V \) is the voltage across the bulb,
- \( R \) is the resistance of the bulb.

### Step 1: Write the equations for each bulb
For the 100W, 60W, and 40W bulbs, we can write the following equations based on the power formula:

1. For the 100W bulb:
\[ P_{100} = \frac{V^2}{R_{100}} \implies R_{100} = \frac{V^2}{100} \]

2. For the 60W bulb:
\[ P_{60} = \frac{V^2}{R_{60}} \implies R_{60} = \frac{V^2}{60} \]

3. For the 40W bulb:
\[ P_{40} = \frac{V^2}{R_{40}} \implies R_{40} = \frac{V^2}{40} \]

### Step 2: Relate the resistances
Now we can relate the resistances \( R_{100}, R_{60}, \) and \( R_{40} \):

From the equations derived above, we have:

\[ R_{100} = \frac{V^2}{100}, \quad R_{60} = \frac{V^2}{60}, \quad R_{40} = \frac{V^2}{40} \]

### Step 3: Take the reciprocal of the resistances
To compare the resistances, we can take the reciprocal of each:

\[ \frac{1}{R_{100}} = \frac{100}{V^2}, \quad \frac{1}{R_{60}} = \frac{60}{V^2}, \quad \frac{1}{R_{40}} = \frac{40}{V^2} \]

### Step 4: Establish the relationship
From the above, we can establish the relationship between the reciprocals of the resistances:

\[ \frac{1}{R_{100}} > \frac{1}{R_{60}} > \frac{1}{R_{40}} \]

This implies:

\[ R_{100} < R_{60} < R_{40} \]

### Conclusion
Thus, the relationship between the resistances of the bulbs is:

\[ R_{100} < R_{60} < R_{40} \]