Electromagnetic Waves

JEE Main level, numerical answer, medium

16 of 165

Question 16 of 165

The electric field of a plane electromagnetic wave, travelling in an unknown nonmagnetic medium is given by, Ey=20sin⁡(3×106x−4.5×1014t)V/mE_{\mathrm{y}}=20 \sin \left(3 \times 10^{6} x-4.5 \times 10^{14} \mathrm{t}\right) \mathrm{V} / \mathrm{m} (where x,tx, \mathrm{t} and other values have S.I. units). The dielectric constant of the medium is ____\_\_\_\_ (speed of light in free space is 3×108 m/s3 \times 10^{8} \mathrm{~m} / \mathrm{s})

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Answer: 4

From the wave equation Ey=20sin⁡(3×106x−4.5×1014t)E_y = 20\sin(3 \times 10^6 x - 4.5 \times 10^{14} t):
Wave number: k=3×106k = 3 \times 10^6 m⁻¹
Angular frequency: ω=4.5×1014\omega = 4.5 \times 10^{14} rad/s
Speed of wave in medium: v=ωk=4.5×10143×106=1.5×108v = \frac{\omega}{k} = \frac{4.5 \times 10^{14}}{3 \times 10^6} = 1.5 \times 10^8 m/s
Speed of light in vacuum: c=3×108c = 3 \times 10^8 m/s
Refractive index: n=cv=3×1081.5×108=2n = \frac{c}{v} = \frac{3 \times 10^8}{1.5 \times 10^8} = 2
For a nonmagnetic medium: n=ϵrn = \sqrt{\epsilon_r}
Dielectric constant: ϵr=n2=4\epsilon_r = n^2 = 4

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