The electric current in a circuit is given by I = 3 sin `omegat + 4 cos omegat.` The rms current is

Options

A. `(5)/sqrt(2)`
B. 5
C. `(4)/sqrt(2)`
D. `(3)/sqrt(2)`

Answer

To find the root mean square (rms) current for the given current expression \( I = 3 \sin(\omega t) + 4 \cos(\omega t) \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Components**:
The current expression can be broken down into two components:
- \( I_1 = 3 \sin(\omega t) \)
- \( I_2 = 4 \cos(\omega t) \)

2. **Calculate the Squares of the Amplitudes**:
Next, we calculate the squares of the amplitudes of these components:
- \( I_1^2 = 3^2 = 9 \)
- \( I_2^2 = 4^2 = 16 \)

3. **Sum the Squares**:
Now, we sum the squares of the amplitudes:
\[ I_1^2 + I_2^2 = 9 + 16 = 25 \]

4. **Calculate the Mean of the Squares**:
To find the mean of the squares, we divide by 2:
\[ \text{Mean} = \frac{I_1^2 + I_2^2}{2} = \frac{25}{2} \]

5. **Take the Square Root**:
Finally, we take the square root of the mean to find the rms current:
\[ I_{\text{rms}} = \sqrt{\frac{I_1^2 + I_2^2}{2}} = \sqrt{\frac{25}{2}} = \frac{5}{\sqrt{2}} \]

### Final Answer:
Thus, the rms current is:
\[ I_{\text{rms}} = \frac{5}{\sqrt{2}} \]