An electromagnetic wave $\vec{E} = \vec{E}_0 e^{i k_0(3x+4y)-\omega t}$, travelling in air is incident on the x – z plane of a glass slab. The glass slab has a refractive index of 1.5. Which one among the following is the correct direction of propagation of the transmitted electromagnetic wave ?
$0.4hat{i} + 0.2sqrt{21}hat{j}$
$0.4hat{i} - 0.2sqrt{21}hat{j}$
$0.4hat{i} + sqrt{21}hat{j}$
$0.4hat{i} - sqrt{21}hat{j}$
Answer is Wrong!
Answer is Right!
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UPSC Geoscientist – 2024
– The incident electromagnetic wave is given by $\vec{E} = \vec{E}_0 e^{i k_0(3x+4y)-\omega t}$. The wave vector $\vec{k}_i$ is the vector part multiplying $\vec{r}=(x,y,z)$ in the exponent (ignoring $k_0$). So, the incident wave vector in air is proportional to $3\hat{i} + 4\hat{j}$. We can write $\vec{k}_i = k_0 (3\hat{i} + 4\hat{j})$.
– The magnitude of the incident wave vector is $|\vec{k}_i| = k_0 \sqrt{3^2 + 4^2} = 5 k_0$. In air, the refractive index $n_1 = 1$, so $|\vec{k}_i| = n_1 (\omega/c) = \omega/c$. This implies $k_0 = \omega/(5c)$.
– The wave is incident on the x-z plane, which is the plane where $y=0$. The normal to this plane is along the $\hat{j}$ direction. Let’s assume the glass slab is in the region $y>0$, and the air is in $y<0$. The incident wave has $k_{iy} = 4k_0 > 0$, so it is travelling towards the interface $y=0$ from the region $y<0$. The normal vector pointing into the glass is $\hat{j}$.
- According to Snell's law for wave vectors, the component of the wave vector parallel to the interface is conserved. The interface is the x-z plane, so the components parallel to the plane are the x and z components.
- Incident wave vector $\vec{k}_i = 3k_0 \hat{i} + 4k_0 \hat{j} + 0\hat{k}$. The parallel component is $k_{ix} \hat{i} + k_{iz} \hat{k} = 3k_0 \hat{i}$.
- The transmitted wave vector in glass is $\vec{k}_t = k_{tx} \hat{i} + k_{ty} \hat{j} + k_{tz} \hat{k}$.
- Conservation of parallel component: $k_{tx} = k_{ix} = 3k_0$ and $k_{tz} = k_{iz} = 0$. So $\vec{k}_t = 3k_0 \hat{i} + k_{ty} \hat{j}$.
- The magnitude of the transmitted wave vector is $|\vec{k}_t| = n_2 (\omega/c) = n_2 n_1^{-1} |\vec{k}_i| = 1.5 \times 1^{-1} \times 5k_0 = 7.5 k_0$.
- $|\vec{k}_t|^2 = k_{tx}^2 + k_{ty}^2 = (3k_0)^2 + k_{ty}^2 = (7.5 k_0)^2$.
- $9k_0^2 + k_{ty}^2 = 56.25 k_0^2$.
- $k_{ty}^2 = (56.25 - 9) k_0^2 = 47.25 k_0^2 = \frac{189}{4} k_0^2$.
- $k_{ty} = \pm \sqrt{\frac{189}{4}} k_0 = \pm \frac{3\sqrt{21}}{2} k_0$.
- Since the wave is transmitted from air ($y<0$) into glass ($y>0$), the y-component of the transmitted wave vector must be positive (pointing into the glass region). So $k_{ty} = \frac{3\sqrt{21}}{2} k_0$.
– The transmitted wave vector is $\vec{k}_t = 3k_0 \hat{i} + \frac{3\sqrt{21}}{2} k_0 \hat{j}$. The direction of propagation is given by the unit vector in the direction of $\vec{k}_t$.
– The direction is proportional to $3\hat{i} + \frac{3\sqrt{21}}{2} \hat{j}$.
– Let’s check the options. Option A is $0.4\hat{i} + 0.2\sqrt{21}\hat{j} = \frac{2}{5}\hat{i} + \frac{\sqrt{21}}{5}\hat{j}$.
– Check if $(3, \frac{3\sqrt{21}}{2})$ is proportional to $(\frac{2}{5}, \frac{\sqrt{21}}{5})$. The ratio of x-components is $3 / (2/5) = 15/2$. The ratio of y-components is $(\frac{3\sqrt{21}}{2}) / (\frac{\sqrt{21}}{5}) = \frac{3\sqrt{21}}{2} \times \frac{5}{\sqrt{21}} = \frac{15}{2}$.
– Since the ratios are equal, the vectors are proportional. The direction of propagation of the transmitted wave is proportional to $0.4\hat{i} + 0.2\sqrt{21}\hat{j}$.