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 RMS (Root Mean Square) current from the given equation of electric current \( I = 3 \sin(\omega t) + 4 \cos(\omega t) \), we will follow these steps:

### Step 1: Identify the given current equation
The current is given as:
\[ I = 3 \sin(\omega t) + 4 \cos(\omega t) \]

### Step 2: Rewrite the equation in standard form
The standard form of an AC current is:
\[ I = I_0 \sin(\omega t + \phi) \]
where \( I_0 \) is the peak current and \( \phi \) is the phase angle. To rewrite the given equation in this form, we need to find \( I_0 \) and \( \phi \).

### Step 3: Find the peak current \( I_0 \)
We can express \( I \) as:
\[ I = R \sin(\omega t + \phi) \]
where \( R \) is the resultant amplitude. We can find \( R \) using the coefficients of sine and cosine:
\[ R = \sqrt{(3)^2 + (4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]
Thus, \( I_0 = R = 5 \).

### Step 4: Calculate the RMS current
The RMS current is given by the formula:
\[ I_{\text{rms}} = \frac{I_0}{\sqrt{2}} \]
Substituting the value of \( I_0 \):
\[ I_{\text{rms}} = \frac{5}{\sqrt{2}} \text{ A} \]

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