Exams › JEE Advanced › Physics
Correct answer: beta = 3*sqrt(14) * 10⁷
The wave vector direction is (1, 2, 3) with magnitude sqrt(1+4+9) = sqrt(14). The angular frequency omega = c * |k| * 10⁷ = 3*10⁸ * sqrt(14) * 10⁷ = 3*sqrt(14) * 10¹⁵ rad/s. So beta = 3*sqrt(14) * 10¹⁵. Wait, let me re-examine the phase: 10⁷(x + 2y + 3z - beta*t). The wave number magnitude is 10⁷ * sqrt(14). So omega = c * k_magnitude = 3*10⁸ * 10⁷ * sqrt(14) = 3*sqrt(14) * 10¹⁵. So beta = 3*sqrt(14) * 10¹⁵. The given option says 3*sqrt(14)*10⁸ which doesn't match dimensionally if beta has units of s⁻¹ / m = rad/m it would be the spatial part... Actually the phase is 10⁷*(x+2y+3z - beta*t) = 10⁷*x + 2*10⁷*y + 3*10⁷*z - 10⁷*beta*t. For EM wave omega = c|k|: 10⁷*beta = c * sqrt(14) * 10⁷, so beta = c*sqrt(14) = 3*10⁸*sqrt(14) m/s. So option (A) should be beta = 3*sqrt(14)*10⁸ m/s, which matches option A. For condition E perpendicular to k: k-direction is (1,2,3), E-amplitude direction is (0,1,b). Dot product: 0*1 + 1*2 + b*3 = 2 + 3b = 0, so b = -2/3. Option (B) b = 2 is incorrect. Average energy density = epsilon0 * E0² (for EM wave in vacuum). E0 = 10⁻³ V/m (magnitude of amplitude vector = sqrt(0+1+b²)*10⁻³ = sqrt(1+4/9)*10⁻³). With b = -2/3: E0 = sqrt(13/9)*10⁻³. u_avg = (1/2)*epsilon0*E0² = (1/2)*epsilon0*(13/9)*10⁻⁶. This doesn't match 6.5*10⁻⁶*epsilon0 (which would require E0² = 13*10⁻⁶... let me try b=2: E0² = (1+4)*10⁻⁶ = 5*10⁻⁶, u_avg = (1/2)*epsilon0*5*10⁻⁶ = 2.5*10⁻⁶*epsilon0. Still not 6.5. If b=3: E0² = (1+9)*10⁻⁶ = 10⁻⁵, u_avg = 5*10⁻⁶ epsilon0. None gives 6.5. The only correct answer is option (A).