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 series circuit consisting of a resistor \( R \) and an inductor \( L \) when a sinusoidal voltage \( V \sin(\omega t) \) is applied, we can follow these steps:

### Step 1: Identify the given parameters
- Voltage: \( V(t) = V_0 \sin(\omega t) \)
- Resistance: \( R \)
- Inductance: \( L \)

### Step 2: Determine the inductive reactance
The inductive reactance \( X_L \) is given by the formula:
\[ X_L = \omega L \]
where \( \omega \) is the angular frequency of the voltage source.

### Step 3: Calculate the impedance of the circuit
In a series circuit with resistance \( R \) and inductance \( L \), the total impedance \( Z \) is given by:
\[ Z = \sqrt{R^2 + X_L^2} \]
Substituting \( X_L \) into the equation, we have:
\[ Z = \sqrt{R^2 + (\omega L)^2} \]

### Step 4: Calculate the amplitude of the current
The amplitude of the current \( I_0 \) can be calculated using Ohm's law for AC circuits:
\[ I_0 = \frac{V_0}{Z} \]
Substituting the expression for \( Z \):
\[ I_0 = \frac{V_0}{\sqrt{R^2 + (\omega L)^2}} \]

### Step 5: Final expression
Thus, the amplitude of the current in the circuit is:
\[ I_0 = \frac{V_0}{\sqrt{R^2 + (\omega L)^2}} \]

### Conclusion
The amplitude of the current in the circuit is given by the formula above.

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