Initial situation

• Polaroids $A$ and $B$ have their pass-axes perpendicular to each other.
• A third polaroid $C$ is placed between them such that its axis bisects the angle between $A$ and $B$.

Therefore, if axis of $A$ is taken at $0^\circ$, then:

• axis of $C$ is at $45^\circ$
• axis of $B$ is at $90^\circ$

Light after polaroid $A$

The incident light is unpolarized with intensity $I_0$.

When unpolarized light passes through an ideal polaroid, transmitted intensity becomes: $I_1 = \frac{I_0}{2}$

Light after polaroid $C$

Now the light is plane polarized along axis of $A$. It passes through polaroid $C$, whose axis makes angle $45^\circ$ with $A$.

By Malus' law: $I_2 = I_1 \cos^2 45^\circ$ $I_2 = \frac{I_0}{2} \cdot \frac{1}{2} = \frac{I_0}{4}$

Light after polaroid $B$

The axis of $B$ is also at $45^\circ$ to the axis of $C$. Again using Malus' law: $I_3 = I_2 \cos^2 45^\circ$ $I_3 = \frac{I_0}{4} \cdot \frac{1}{2} = \frac{I_0}{8}$

Final answer

The transmitted intensity after passing through polaroid $B$ is: $\boxed{\frac{I_0}{8}}$

Option check

• A: $\frac{I_0}{4}$ ❌
• B: $\frac{I_0}{8}$ ✅
• C: Zero ❌
• D: $\frac{I_0}{2}$ ❌