Electromagnetic Waves Wave Equation Shortcut

 

❓ Question

The electric field of an EM wave is:

$$E = 2\sin\left(2\times10^{15}t - 10^7x\right)$$

Find the refractive index of the medium.

Options:

1. $\frac{3}{2}$
2. $2$
   
3. $\frac{5}{3}$
4. $\frac{4}{3}$​

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✍️ Short Explanation

Compare the given equation with the standard EM wave form:

$$E = E_0\sin(\omega t-kx)$$

where

$$\omega = 2\times10^{15}\ \text{rad/s}$$

and

$$k = 10^7\ \text{rad/m}$$

The wave speed is:

$$v=\frac{\omega}{k}$$

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🔷 Step 1 — Calculate Speed of Wave

$$v=\frac{2\times10^{15}}{10^7}$$
$$v=2\times10^8\ \text{m/s}$$

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🔷 Step 2 — Calculate Refractive Index

$$n=\frac{c}{v}$$

where

$$c=3\times10^8\ \text{m/s}$$

Thus,

$$n=\frac{3\times10^8}{2\times10^8}$$
$$n=\frac{3}{2}$$

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✅ Final Answer

$$\boxed{\frac{3}{2}}$$

Option (1)

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🔷 JEE Shortcut

For a wave:

$$E=E_0\sin(\omega t-kx)$$

directly use:

$$n=\frac{ck}{\omega}$$

So,

$$n=\frac{(3\times10^8)(10^7)}{2\times10^{15}} =\frac{3}{2}$$
$$\boxed{\frac{3}{2}}$$