JEE Main 2021 Physics Question Paper with Answer and Solution

(A) Let the thickness of the columns be $t$. The number of waves in vacuum is $N_v = t / \lambda$ and the number of waves in air is $N_a = t / \lambda_a$,where $\lambda_a = \lambda / \mu_{air}$.
Given that the difference in the number of waves is $1$,we have $N_v - N_a = 1$.
Substituting the expressions: $t / \lambda - t / (\lambda / \mu_{air}) = 1$.
$t / \lambda - (t \cdot \mu_{air}) / \lambda = 1$.
$(t / \lambda) (1 - \mu_{air}) = 1$.
Since $\mu_{air} = 1.0003$,the magnitude of the difference is $(t / \lambda) (\mu_{air} - 1) = 1$.
$t = \lambda / (\mu_{air} - 1) = 6000 \times 10^{-10} \, m / (1.0003 - 1) = 6000 \times 10^{-10} / 0.0003$.
$t = 6000 \times 10^{-10} / (3 \times 10^{-4}) = 2000 \times 10^{-6} \, m = 2 \times 10^{-3} \, m = 2 \, mm$.

(B) According to Faraday's law of electromagnetic induction,the induced electromotive force (emf) is given by the magnitude of the rate of change of magnetic flux:
$|\epsilon| = \left| \frac{d\phi_{B}}{dt} \right|$
Given $\phi_{B}(t) = 10t^{2} + 20t$ (in $mWb$),
$|\epsilon| = \frac{d}{dt}(10t^{2} + 20t) = 20t + 20$ (in $mV$)
Using Ohm's law,the induced current $i$ is given by:
$|i| = \frac{|\epsilon|}{R} = \frac{20t + 20}{2} = 10t + 10$ (in $mA$)
At $t = 5\,s$:
$|i| = 10(5) + 10 = 50 + 10 = 60\,mA$.

(C) Given the ratio of intensities of two coherent sources is $\frac{I_1}{I_2} = 2x$.
We know that $I_{\max} = (\sqrt{I_1} + \sqrt{I_2})^2$ and $I_{\min} = (\sqrt{I_1} - \sqrt{I_2})^2$.
The expression to evaluate is $V = \frac{I_{\max} - I_{\min}}{I_{\max} + I_{\min}}$.
Substituting the expressions for $I_{\max}$ and $I_{\min}$:
$V = \frac{(\sqrt{I_1} + \sqrt{I_2})^2 - (\sqrt{I_1} - \sqrt{I_2})^2}{(\sqrt{I_1} + \sqrt{I_2})^2 + (\sqrt{I_1} - \sqrt{I_2})^2}$.
Using the algebraic identities $(a+b)^2 - (a-b)^2 = 4ab$ and $(a+b)^2 + (a-b)^2 = 2(a^2 + b^2)$:
$V = \frac{4\sqrt{I_1 I_2}}{2(I_1 + I_2)} = \frac{2\sqrt{I_1 I_2}}{I_1 + I_2}$.
Dividing the numerator and denominator by $I_2$:
$V = \frac{2\sqrt{I_1/I_2}}{I_1/I_2 + 1}$.
Substituting $\frac{I_1}{I_2} = 2x$:
$V = \frac{2\sqrt{2x}}{2x + 1}$.