An AC source producing `emf`
`epsilon = epsilon_0 [cos(100 pi s^(-1)) t + cos (500 pi s^(-1))t]`
is connected in series with a capacitor and a resistor. The steady-state current in the circuit is found to be
`I = i_1 cos [(100 pi s^(-1) t + varphi_1] + i_2 cos [(500 pi s^(-1))t +phi_2]`.

Options

A. `i_1 gt i_2`
B. `i_1 = i_2`
C. `i_1 lt i_2`
D. The information is insufficient to find the relation between `i_1 and i_2`.

Answer

To solve the problem, we need to analyze the given AC source and the circuit components involved. We are given the EMF of the AC source and the steady-state current in the circuit. Let's break down the solution step by step.

### Step 1: Understand the EMF of the AC source
The EMF of the AC source is given as:
\[ \epsilon = \epsilon_0 \left[ \cos(100 \pi t) + \cos(500 \pi t) \right] \]
This indicates that the source has two frequency components: one at \(100 \pi \, \text{s}^{-1}\) and another at \(500 \pi \, \text{s}^{-1}\).

### Step 2: Identify the current components
The steady-state current in the circuit is given as:
\[ I = I_1 \cos(100 \pi t + \varphi_1) + I_2 \cos(500 \pi t + \varphi_2) \]
Here, \(I_1\) and \(I_2\) are the amplitudes of the current corresponding to the two frequency components.

### Step 3: Relate current to charge on the capacitor
The charge \(Q\) on the capacitor is given by:
\[ Q = C \epsilon \]
Substituting the expression for \(\epsilon\):
\[ Q = C \epsilon_0 \left[ \cos(100 \pi t) + \cos(500 \pi t) \right] \]

### Step 4: Differentiate charge to find current
The current \(I\) is the rate of change of charge with respect to time:
\[ I = \frac{dQ}{dt} \]
Differentiating \(Q\):
\[ I = \frac{d}{dt} \left( C \epsilon_0 \left[ \cos(100 \pi t) + \cos(500 \pi t) \right] \right) \]
Using the chain rule, we get:
\[ I = C \epsilon_0 \left[ -100 \pi \sin(100 \pi t) - 500 \pi \sin(500 \pi t) \right] \]

### Step 5: Express current in cosine form
We can rewrite the sine functions in terms of cosine:
\[ I = C \epsilon_0 \left[ 100 \pi \sin(100 \pi t) + 500 \pi \sin(500 \pi t) \right] \]
Using the identity \(\sin(x) = \cos\left(x - \frac{\pi}{2}\right)\), we can express the current as:
\[ I = C \epsilon_0 \left[ 100 \pi \cos(100 \pi t - \frac{\pi}{2}) + 500 \pi \cos(500 \pi t - \frac{\pi}{2}) \right] \]

### Step 6: Compare amplitudes
From the expression for current, we can identify:
\[ I_1 = C \epsilon_0 \cdot 100 \pi \]
\[ I_2 = C \epsilon_0 \cdot 500 \pi \]

### Step 7: Determine the relationship between \(I_1\) and \(I_2\)
Now we can compare \(I_1\) and \(I_2\):
\[ I_2 = 5 I_1 \]
This implies that:
\[ I_2 > I_1 \]

### Conclusion
Thus, the correct option is that \(I_2\) is greater than \(I_1\).