two resistors R and 2R are connected in series in an electric circuit. The thermal energy developed in R and 2R are in the ratio

Options

A. (a)1:2
B. (b)2:1
C. (c )1:4
D. (d)4:1

Answer

To solve the problem of finding the ratio of thermal energy developed in resistors R and 2R connected in series, we can follow these steps:

### Step 1: Understand the Circuit Configuration
In this scenario, we have two resistors, R and 2R, connected in series. When resistors are connected in series, the same current flows through both resistors.

### Step 2: Write the Expression for Power
The power (P) developed in a resistor can be expressed using the formula:
\[ P = I^2 R \]
where \( I \) is the current flowing through the resistor and \( R \) is the resistance.

### Step 3: Calculate the Power in Each Resistor
For resistor R:
\[ P_R = I^2 R \]

For resistor 2R:
\[ P_{2R} = I^2 (2R) = 2I^2 R \]

### Step 4: Find the Ratio of Power Developed
Now, we can find the ratio of the power developed in R to that in 2R:
\[ \text{Ratio} = \frac{P_R}{P_{2R}} = \frac{I^2 R}{2I^2 R} \]
This simplifies to:
\[ \text{Ratio} = \frac{1}{2} \]

### Step 5: Relate Power to Thermal Energy
Since thermal energy developed (Q) in a resistor over time (t) is given by:
\[ Q = P \cdot t \]
The ratio of thermal energy developed in R and 2R will be the same as the ratio of their powers, as time is constant for both resistors.

### Final Answer
Thus, the ratio of thermal energy developed in R and 2R is:
\[ \text{Ratio} = 1 : 2 \]