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Question:

An electromagnetic wave travelling in the x-direction has a frequency of 2 × 10¹⁴ Hz and an electric field amplitude of 27 V/m. From the options given below, which one describes the magnetic field for this wave?

→B(x,t)=(3 × 10⁻⁸ T)^j sin[2π(1.5 × 10⁷x − 7 × 10¹⁴t)]

→B(x,t)=(9 × 10⁻⁸ T)^i sin[2π(1.5 × 10⁷x − 7 × 10¹⁴t)]

→B(x,t)=(9 × 10⁻⁸ T)^k sin[2π(1.5 × 10⁷x − 7 × 10¹⁴t)]

→B(x,t)=(9 × 10⁻⁸ T)^j sin[(1.5 × 10⁷x − 7 × 10¹⁴t)]

Solution:

Using Maxwell's equation E × B = c, where c is the velocity of light, E and B are amplitudes of electric and magnetic fields. Given E = 27 N/C, the amplitude of the magnetic field is B = 27/(3 × 10⁸) = 9 × 10⁻⁸ T.

The frequency of the wave is given as f = 2 × 10¹⁴ Hz; its angular frequency is ω = 2πf = 2π × 2 × 10¹⁴.

The equation of the magnetic field can be given by B sin(kx − ωt) = B sin(2π(x/λ − ft)) = 9 × 10⁻⁸ sin(2π(x/λ − 2 × 10¹⁴t)) (^j or ^k)

Since the direction of the electromagnetic wave is given as the x-axis, the direction of the magnetic field will be perpendicular to it. Therefore, options B and C are possible answers. However, in option B, the angular frequency is not calculated correctly. Option C correctly represents both the amplitude and direction of the magnetic field, accounting for the wave's propagation in the x-direction.