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Two bulbs consume the same power when op...

Two bulbs consume the same power when operated at 200 V and 300 V respectively. When these bulbs are connected in series across a DC source of 400 V, then the ratio of power consumed across them is

A

`(2)/(3)`

B

`(3)/(2)`

C

`(4)/(9)`

D

`(9)/(4)`

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we need to find the ratio of power consumed by two bulbs when connected in series across a DC source of 400 V. The bulbs consume the same power when operated at 200 V and 300 V respectively. ### Step-by-Step Solution: 1. **Understanding Power Consumption**: The power consumed by an electrical device can be expressed using the formula: \[ P = \frac{V^2}{R} \] where \( P \) is the power, \( V \) is the voltage, and \( R \) is the resistance. 2. **Setting Up the Equations**: For the first bulb (operating at 200 V): \[ P_1 = \frac{200^2}{R_1} \] For the second bulb (operating at 300 V): \[ P_2 = \frac{300^2}{R_2} \] Since both bulbs consume the same power: \[ \frac{200^2}{R_1} = \frac{300^2}{R_2} \] 3. **Finding the Resistance Ratio**: Rearranging the equation gives: \[ \frac{R_1}{R_2} = \frac{200^2}{300^2} = \left(\frac{200}{300}\right)^2 = \left(\frac{2}{3}\right)^2 = \frac{4}{9} \] Thus, we have: \[ R_1 : R_2 = 4 : 9 \] 4. **Connecting Bulbs in Series**: When the bulbs are connected in series, the same current \( I \) flows through both. The total voltage across the series connection is 400 V, and the voltage across each bulb can be expressed as: \[ V_1 = I R_1 \quad \text{and} \quad V_2 = I R_2 \] The total voltage is: \[ V_1 + V_2 = 400 \quad \Rightarrow \quad I R_1 + I R_2 = 400 \quad \Rightarrow \quad I(R_1 + R_2) = 400 \] 5. **Calculating Power in Series**: The power consumed by each bulb can be expressed as: \[ P_1 = I^2 R_1 \quad \text{and} \quad P_2 = I^2 R_2 \] The ratio of the power consumed by the two bulbs is: \[ \frac{P_1}{P_2} = \frac{I^2 R_1}{I^2 R_2} = \frac{R_1}{R_2} \] Since we found \( R_1 : R_2 = 4 : 9 \), we have: \[ \frac{P_1}{P_2} = \frac{4}{9} \] ### Final Answer: The ratio of power consumed across the two bulbs when connected in series across a 400 V source is: \[ \frac{P_1}{P_2} = \frac{4}{9} \]
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