A current is made of two components a `dc` component `i_(1) = 3A` and an `ac` component `i_(2) = 4 sqrt(2) sin omega t`. Find the reading of hot wire ammeter?

Options

A. 4amp
B. `4sqrt2amp`
C. `(3+4sqrt2)amp`
D. 5amp

Answer

To find the reading of a hot wire ammeter for the given current components, we will follow these steps:

### Step 1: Identify the Components of Current
The total current \( i(t) \) consists of a DC component \( i_1 \) and an AC component \( i_2 \):
- \( i_1 = 3 \, \text{A} \) (DC component)
- \( i_2 = 4\sqrt{2} \sin(\omega t) \) (AC component)

### Step 2: Write the Total Current Expression
The total current can be expressed as:
\[ i(t) = i_1 + i_2 = 3 + 4\sqrt{2} \sin(\omega t) \]

### Step 3: Calculate the RMS Value of the Current
The reading of the hot wire ammeter is proportional to the root mean square (RMS) value of the total current. The RMS value of a current that has both DC and AC components can be calculated using the formula:
\[ I_{\text{rms}} = \sqrt{I_{\text{dc}}^2 + I_{\text{ac, rms}}^2} \]
Where:
- \( I_{\text{dc}} = i_1 = 3 \, \text{A} \)
- \( I_{\text{ac, rms}} \) is the RMS value of the AC component.

### Step 4: Calculate the RMS Value of the AC Component
The RMS value of the AC component \( i_2 = 4\sqrt{2} \sin(\omega t) \) is given by:
\[ I_{\text{ac, rms}} = \frac{I_{\text{peak}}}{\sqrt{2}} = \frac{4\sqrt{2}}{\sqrt{2}} = 4 \, \text{A} \]

### Step 5: Substitute Values into the RMS Formula
Now, substitute the values into the RMS formula:
\[ I_{\text{rms}} = \sqrt{(3)^2 + (4)^2} \]
\[ I_{\text{rms}} = \sqrt{9 + 16} = \sqrt{25} = 5 \, \text{A} \]

### Step 6: Conclusion
The reading of the hot wire ammeter, which measures the RMS value of the current, is:
\[ \text{Reading of hot wire ammeter} = 5 \, \text{A} \]

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