1. Set up the polarization geometry

Since beams $A$ and $B$ are plane polarized in mutually perpendicular planes, let the transmission axis of the polaroid initially be aligned with beam $A$.

Then initially:

• Beam $A$ has maximum transmitted intensity.
• Beam $B$ has zero transmitted intensity.

This matches the statement in the question.

2. Apply Malus' law after rotating the polaroid by $30^\circ$

After rotating the polaroid by $30^\circ$ from the initial position:

• The angle between beam $A$'s polarization and the polaroid axis is $30^\circ$.
• Since beam $B$ is polarized perpendicular to $A$, the angle between beam $B$'s polarization and the polaroid axis is $60^\circ$.

By Malus' law, transmitted intensity is $I = I_0 \cos^2 \theta.$

So,

For beam $A$: $I_A' = I_A \cos^2 30^\circ = I_A \cdot \frac{3}{4}.$

For beam $B$: $I_B' = I_B \cos^2 60^\circ = I_B \cdot \frac{1}{4}.$

3. Use the condition of equal brightness

Given that after rotation, the two beams appear equally bright: $I_A' = I_B'.$

Thus, $I_A \cdot \frac{3}{4} = I_B \cdot \frac{1}{4}.$

Multiply both sides by $4$: $3I_A = I_B.$

Hence, $\frac{I_A}{I_B} = \frac{1}{3}.$

4. Match with the options

$\frac{I_A}{I_B} = \frac{1}{3}$ which corresponds to Option D.

5. Comparison with stored answer

Stored correct answer: D

Derived answer: D

So, the derived answer agrees with the stored answer.