Which of the following diagrams represent the veriation of electric field vector with time for a circularly polarised light

Options

A.
B.
C.
D.

Explanation

To solve the question regarding which diagram represents the variation of the electric field vector with time for circularly polarized light, we can follow these steps:

### Step 1: Understand Circularly Polarized Light
Circularly polarized light consists of two perpendicular electric field components that are out of phase by 90 degrees (or π/2 radians). These components can be represented as:
- \( E_x = E_0 \sin(\omega t) \)
- \( E_y = E_0 \sin(\omega t + \frac{\pi}{2}) \)

### Step 2: Express the Second Component
Using the trigonometric identity, we can express the second component \( E_y \):
- \( E_y = E_0 \sin(\omega t + \frac{\pi}{2}) = E_0 \cos(\omega t) \)

### Step 3: Determine the Resultant Electric Field
The resultant electric field vector can be expressed as:
- \( \vec{E} = E_x \hat{i} + E_y \hat{j} \)
Substituting the expressions we have:
- \( \vec{E} = E_0 \sin(\omega t) \hat{i} + E_0 \cos(\omega t) \hat{j} \)

### Step 4: Calculate the Magnitude of the Electric Field
The magnitude of the electric field can be calculated as:
- \( |\vec{E}| = \sqrt{E_x^2 + E_y^2} \)
Substituting the expressions for \( E_x \) and \( E_y \):
- \( |\vec{E}| = \sqrt{(E_0 \sin(\omega t))^2 + (E_0 \cos(\omega t))^2} \)
- \( |\vec{E}| = E_0 \sqrt{\sin^2(\omega t) + \cos^2(\omega t)} \)

### Step 5: Simplify the Magnitude
Using the Pythagorean identity \( \sin^2(\theta) + \cos^2(\theta) = 1 \):
- \( |\vec{E}| = E_0 \)

### Step 6: Conclusion
The magnitude of the electric field remains constant over time, which indicates that the electric field vector rotates in a circular motion in the plane perpendicular to the direction of propagation of the light. Thus, the correct diagram representing this behavior will show a constant magnitude with a rotating direction.

### Final Answer
Based on the analysis, the correct option is **Option A**, which represents the constant magnitude of the electric field vector with time for circularly polarized light.

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