A sinusoidal voltage Vsin(at) is applied across a series combination of resistance R and inductor L. The amplitude of the current in the circuit is

Options

A. `V_(0)/(sqrt(R^(2)+omega^(2)L^(2)))`
B. `(V_(0))/(sqrt(R^(2)-omega^(2)L^(2)))`
C. `(V_(0))/(sqrt(R^(2)+omega^(2)L^(2)))sinomegat`
D. VsqrtR`

Explanation

To find the amplitude of the current in a circuit with a sinusoidal voltage applied across a series combination of resistance (R) and an inductor (L), we can follow these steps:

### Step 1: Identify the given parameters
We have:
- A sinusoidal voltage \( V(t) = V_0 \sin(\omega t) \)
- Resistance \( R \)
- Inductance \( L \)

### Step 2: Understand the concept of impedance
In an R-L series circuit, the total impedance \( Z \) can be calculated using the formula:
\[ Z = \sqrt{R^2 + X_L^2} \]
where \( X_L \) is the inductive reactance.

### Step 3: Calculate the inductive reactance
The inductive reactance \( X_L \) is given by:
\[ X_L = \omega L \]
where \( \omega \) is the angular frequency of the sinusoidal voltage.

### Step 4: Substitute \( X_L \) into the impedance formula
Now substituting \( X_L \) into the impedance formula:
\[ Z = \sqrt{R^2 + (\omega L)^2} \]

### Step 5: Calculate the amplitude of the current
The amplitude of the current \( I_0 \) in the circuit can be calculated using Ohm's law for AC circuits:
\[ I_0 = \frac{V_0}{Z} \]
Substituting the expression for \( Z \):
\[ I_0 = \frac{V_0}{\sqrt{R^2 + (\omega L)^2}} \]

### Final Answer
Thus, the amplitude of the current in the circuit is:
\[ I_0 = \frac{V_0}{\sqrt{R^2 + \omega^2 L^2}} \]