A heating coil is rated `100 W , 200 V`. The coil is cut in half and two pieces are joined in parallel to the same source . Now what is the energy `( "in" xx 10^(2) J)` liberated per second?

Explanation

To solve the problem step by step, let's break it down:

### Step 1: Understand the initial conditions
The heating coil is rated at 100 W and 200 V. This means that when connected to a 200 V source, it consumes 100 W of power.

### Step 2: Calculate the resistance of the original coil
Using the formula for power:
\[ P = \frac{V^2}{R} \]
We can rearrange this to find the resistance \( R \):
\[ R = \frac{V^2}{P} \]
Substituting the given values:
\[ R = \frac{(200)^2}{100} = \frac{40000}{100} = 400 \, \Omega \]

### Step 3: Determine the resistance of each half of the coil
When the coil is cut in half, the resistance of each half will be:
\[ R_{half} = \frac{R}{2} = \frac{400 \, \Omega}{2} = 200 \, \Omega \]

### Step 4: Calculate the equivalent resistance when the two halves are connected in parallel
For two resistors \( R_1 \) and \( R_2 \) in parallel, the equivalent resistance \( R_{eq} \) is given by:
\[ \frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} \]
Here, both resistances are equal (200 Ω):
\[ \frac{1}{R_{eq}} = \frac{1}{200} + \frac{1}{200} = \frac{2}{200} = \frac{1}{100} \]
Thus,
\[ R_{eq} = 100 \, \Omega \]

### Step 5: Calculate the new power consumed by the parallel arrangement
Using the power formula again:
\[ P_{new} = \frac{V^2}{R_{eq}} \]
Substituting the values:
\[ P_{new} = \frac{(200)^2}{100} = \frac{40000}{100} = 400 \, W \]

### Step 6: Determine the energy liberated per second
Since power is defined as energy per unit time (1 Watt = 1 Joule/second), the energy liberated per second is equal to the power:
\[ \text{Energy liberated per second} = 400 \, J \]

### Step 7: Express the answer in the required format
The question asks for the energy liberated in the form of \( "in" \times 10^2 J \):
\[ 400 \, J = 4 \times 10^2 \, J \]

### Final Answer
The energy liberated per second is \( 4 \times 10^2 \, J \).

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