Question

For the electric field of an electromagnetic wave given below, which of the following statements is correct?
\[ \vec{E} = \hat{x}\, E_0 \cos(\omega t) + \hat{y}\, 2E_0 \cos\left(\omega t + \frac{\pi}{2}\right) \]

• A. The electric field is linearly polarised with slope 2.
• B. The electric field is circularly polarised with radius \(E_0\).
• C. The electric field is elliptically polarised with a ratio of major to minor axis being 2.
• D. The electric field is unpolarised with the two components being phase shifted by \(\pi/2\).

Hint:

Write \(E_y=2E_0\cos(\omega t+\pi/2)=-2E_0\sin\omega t\) and eliminate \(t\) between \(E_x\) and \(E_y\).
The result is an ellipse equation with semi-axes \(E_0\) and \(2E_0\).

Answer 1

The Correct Option is C

Step 1: Use complex phasor amplitudes instead of trig identities.
Write each component as the real part of a complex phasor times $e^{-i\omega t}$: $E_x = \text{Re}[\tilde E_x e^{-i\omega t}]$ with $\tilde E_x = E_0$, and $E_y = \text{Re}[\tilde E_y e^{-i\omega t}]$ with $\tilde E_y = 2E_0\, e^{i\pi/2} = 2iE_0$, since $e^{i\pi/2} = i$.

Step 2: Read the polarisation straight off the ratio of the two phasors.
Form the ratio $\tilde E_y/\tilde E_x = 2iE_0/E_0 = 2i$, a purely imaginary number. This one ratio tells us everything: if it were real, the two components would rise and fall together (or oppositely) and the wave would be linearly polarised; being purely imaginary means the components are exactly a quarter cycle out of step, which is the signature of circular or elliptical polarisation, never linear.

Step 3: Decide between circular and elliptical using the size of the ratio.
$$|\tilde E_y/\tilde E_x| = 2 \ne 1$$
A purely imaginary ratio with magnitude exactly 1 would mean equal amplitudes tracing a circle. Here the magnitude is 2, not 1, so the amplitudes along $x$ and $y$ are unequal and the tip of the field vector instead traces an ellipse whose axes sit along $x$ and $y$, with the axis ratio equal to this same magnitude, 2.

Step 4: Final Answer.
The purely imaginary phasor ratio $2i$ directly gives elliptical polarisation with major to minor axis ratio 2, confirming option (C) without ever writing out sine and cosine separately. \[ \boxed{\text{Option (C): elliptical, axis ratio } 2} \]