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An electromagnetic wave propagates in a medium, whose electric field vector is given as E=200sin⁡(18×108t+8x)E = 200\sin (18\times 10^8t + 8x) V m−1^{-1}. Refractive index of medium and amplitude of magnetic field will be

Electromagnetic Waves
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1

μ=32;B0=23×10−6\mu = \frac{3}{2}; B_0 = \frac{2}{3}\times 10^{-6} T

2

μ=43;B0=23×10−6\mu = \frac{4}{3}; B_0 = \frac{2}{3}\times 10^{-6} T

3

μ=32;B0=89×10−6\mu = \frac{3}{2}; B_0 = \frac{8}{9}\times 10^{-6} T

4

μ=43;B0=89×10−6\mu = \frac{4}{3}; B_0 = \frac{8}{9}\times 10^{-6} T

Solution:

The given electric field is E=200sin⁡(18×108t+8x)E = 200\sin (18\times 10^8t + 8x). This equation is in the form E=E0sin⁡(ωt+kx)E = E_0 \sin(\omega t + kx).

Comparing the given equation with the standard form, we get:

Amplitude of electric field, E0=200E_0 = 200 V/m

Angular frequency, ω=18×108\omega = 18 \times 10^8 rad/s

Wave number, k=8k = 8 rad/m

The speed of the electromagnetic wave in the medium is given by v=ωkv = \frac{\omega}{k}.

v=18×1088=94×108v = \frac{18 \times 10^8}{8} = \frac{9}{4} \times 10^8 m/s.

The refractive index of the medium, μ\mu, is the ratio of the speed of light in vacuum (cc) to the speed of light in the medium (vv). The speed of light in vacuum is approximately c=3×108c = 3 \times 10^8 m/s.

μ=cv=3×10894×108=3×49=129=43\mu = \frac{c}{v} = \frac{3 \times 10^8}{\frac{9}{4} \times 10^8} = \frac{3 \times 4}{9} = \frac{12}{9} = \frac{4}{3}.

The amplitude of the magnetic field, B0B_0, is related to the amplitude of the electric field, E0E_0, and the speed of the wave in the medium, vv, by the relation B0=E0vB_0 = \frac{E_0}{v}.

B0=20094×108=200×49×108=8009×108=8009×10−8=8×1009×10−8=89×102×10−8=89×10−6B_0 = \frac{200}{\frac{9}{4} \times 10^8} = \frac{200 \times 4}{9 \times 10^8} = \frac{800}{9 \times 10^8} = \frac{800}{9} \times 10^{-8} = \frac{8 \times 100}{9} \times 10^{-8} = \frac{8}{9} \times 10^2 \times 10^{-8} = \frac{8}{9} \times 10^{-6} T.

Thus, the refractive index of the medium is 43\frac{4}{3} and the amplitude of the magnetic field is 89×10−6\frac{8}{9} \times 10^{-6} T.

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