An dielectric current has both DC and AC components . DC component of 8 A and AC component is given as I = `6sin`omega`t. So (rms)(I) value of resultant current is

Options

A. 8.05 A
B. 9.05 A
C. 11.58 A
D. 13.58 A

Explanation

To find the RMS value of the resultant current that has both DC and AC components, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Components**:
- The DC component is given as \( I_{DC} = 8 \, \text{A} \).
- The AC component is given as \( I_{AC} = 6 \sin(\omega t) \).

2. **Write the Expression for Resultant Current**:
- The resultant current \( I \) can be expressed as:
\[ I = I_{DC} + I_{AC} = 8 + 6 \sin(\omega t) \]

3. **Square the Resultant Current**:
- To find the RMS value, we first need to square the resultant current:
\[ I^2 = (8 + 6 \sin(\omega t))^2 \]
- Expanding this using the formula \( (a + b)^2 = a^2 + 2ab + b^2 \):
\[ I^2 = 8^2 + 2 \cdot 8 \cdot 6 \sin(\omega t) + (6 \sin(\omega t))^2 \]
- This simplifies to:
\[ I^2 = 64 + 96 \sin(\omega t) + 36 \sin^2(\omega t) \]

4. **Calculate the Mean Value of \( I^2 \)**:
- The mean value of \( I^2 \) over one complete cycle is calculated as follows:
- The mean of a constant (64) is \( 64 \).
- The mean of \( 36 \sin^2(\omega t) \) is \( 36 \cdot \frac{1}{2} = 18 \) (since the average value of \( \sin^2 \) over a cycle is \( \frac{1}{2} \)).
- The mean of \( 96 \sin(\omega t) \) is \( 0 \) (since the average value of \( \sin \) over a cycle is \( 0 \)).
- Therefore, the mean value of \( I^2 \) is:
\[ \text{Mean}(I^2) = 64 + 18 + 0 = 82 \]

5. **Calculate the RMS Value**:
- The RMS value \( I_{rms} \) is the square root of the mean value of \( I^2 \):
\[ I_{rms} = \sqrt{\text{Mean}(I^2)} = \sqrt{82} \]
- Calculating this gives:
\[ I_{rms} \approx 9.055 \, \text{A} \]

### Final Answer:
The RMS value of the resultant current is approximately \( 9.05 \, \text{A} \).

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