10 kJ/s of heat is produced when a DC flows through a `100 Omega` resistor. Which of the following ac produces nearly same heat per sec through the same resistor ?

Options

A. `l = 15 sin omega t`
B. `l = 45 sin omega t`
C. `l = 30 sin omega t`
D. `l=10 sin omega t`

Answer

To solve the problem step by step, we need to determine the AC current that produces the same amount of heat per second in a 100 Ω resistor as the given DC current.

### Step-by-Step Solution:

1. **Understanding the Heat Produced by DC:**
The heat produced (H) in a resistor when a current (I) flows through it for time (t) is given by the formula:
\[ H = I^2 R t \]
In this case, we are given that \( H = 10 \, \text{kJ/s} = 10,000 \, \text{J/s} \) and \( R = 100 \, \Omega \). Since we want to find the heat produced per second, we can set \( t = 1 \, \text{s} \).

2. **Substituting the Values:**
We can rearrange the formula to solve for the current \( I \):
\[ 10,000 = I^2 \times 100 \times 1 \]
Simplifying this gives:
\[ I^2 = \frac{10,000}{100} = 100 \]

3. **Calculating the DC Current:**
Taking the square root of both sides, we find:
\[ I = \sqrt{100} = 10 \, \text{A} \]
So, the DC current is 10 A.

4. **Finding the Equivalent AC Current:**
For AC, the effective (RMS) current \( I_{\text{rms}} \) is related to the peak current \( I_0 \) by the equation:
\[ I_{\text{rms}} = \frac{I_0}{\sqrt{2}} \]
Since we want the heat produced by AC to be the same as that produced by the DC current, we can set:
\[ H = I_{\text{rms}}^2 R t \]
We already know \( H = 10,000 \, \text{J/s} \) and \( R = 100 \, \Omega \).

5. **Setting Up the Equation for AC:**
Substituting the known values into the equation:
\[ 10,000 = I_{\text{rms}}^2 \times 100 \times 1 \]
Rearranging gives:
\[ I_{\text{rms}}^2 = \frac{10,000}{100} = 100 \]

6. **Calculating the RMS Current:**
Taking the square root:
\[ I_{\text{rms}} = \sqrt{100} = 10 \, \text{A} \]

7. **Finding the Peak Current:**
To find the peak current \( I_0 \):
\[ I_0 = I_{\text{rms}} \times \sqrt{2} = 10 \times \sqrt{2} \approx 10 \times 1.414 = 14.14 \, \text{A} \]

8. **Choosing the Closest Option:**
The options provided are 15 A, 45 A, 30 A, and 10 A. The closest value to 14.14 A is 15 A.

### Final Answer:
The AC current that produces nearly the same heat per second through the same resistor is approximately **15 A**.