Two light beams fall on a transparent material block at point 1 and 2 with angle theta_1$\theta_{1}$ and theta_2$\theta_{2}$ , respectively, as shown in figure. After refraction, the beams intersect at point 3 which is exactly on the interface at other end of the block. Given: the distance between 1 and 2, d = 4sqrt3 mathrm~cm$d = 4\sqrt{3} \mathrm{~cm}$ and theta_1 = theta_2 = cos^-1left(fracn_22n_1right)$\theta_{1} = \theta_{2} = \cos^{-1}\left(\frac{n_{2}}{2n_{1}}\right)$ , where refractive index of the block n_2 >$n_{2} >$ refractive index of the outside medium n_1$n_{1}$ , then the thickness of the block is ______ cm.

### Related Formula mu = fracsinleft(fracA + delta_m2right)sinleft(fracA2right)$$\mu = \frac{\sin\left(\frac{A + \delta_m}{2}\right)}{\sin\left(\frac{A}{2}\right)}$$ ### Core Logic Given that the refractive index mu = cotleft(fracA2right)$\mu = \cot\left(\frac{A}{2}\right)$. Substituting this into the prism formula: cotleft(fracA2right) = fracsinleft(fracA + delta_m2right)sinleft(fracA2right)$$\cot\left(\frac{A}{2}\right) = \frac{\sin\left(\frac{A + \delta_m}{2}\right)}{\sin\left(\frac{A}{2}\right)}$$ fraccosleft(fracA2right)sinleft(fracA2right) = fracsinleft(fracA + delta_m2right)sinleft(fracA2right)$$\frac{\cos\left(\frac{A}{2}\right)}{\sin\left(\frac{A}{2}\right)} = \frac{\sin\left(\frac{A + \delta_m}{2}\right)}{\sin\left(\frac{A}{2}\right)}$$ Equating the numerators: cosleft(fracA2right) = sinleft(fracA + delta_m2right)$$\cos\left(\frac{A}{2}\right) = \sin\left(\frac{A + \delta_m}{2}\right)$$ We can rewrite cosine in terms of sine: sinleft(fracpi2 - fracA2right) = sinleft(fracA + delta_m2right)$$\sin\left(\frac{\pi}{2} - \frac{A}{2}\right) = \sin\left(\frac{A + \delta_m}{2}\right)$$ ### Step 2: Solve for Deviation Comparing the angles inside the sine functions: fracpi2 - fracA2 = fracA2 + fracdelta_m2$$\frac{\pi}{2} - \frac{A}{2} = \frac{A}{2} + \frac{\delta_m}{2}$$ Multiply the entire equation by 2: pi - A = A + delta_m$$\pi - A = A + \delta_m$$ delta_m = pi - 2A$$\delta_m = \pi - 2A$$ Converting radians to degrees: delta_m = 180^circ - 2A$$\delta_m = 180^{\circ} - 2A$$ ### Pattern Recognition Whenever refractive index mu = cot(A/2)$\mu = \cot(A/2)$, the relation sin(90^circ - A/2)$\sin(90^\circ - A/2)$ strictly matches the prism sine equation, meaning minimum deviation delta_m$\delta_m$ is always 180^circ - 2A$180^\circ - 2A$. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 12 Physics: Ray Optics And Optical Instruments