The magnetic field of a plane electromagnetic wave is given by B = 8 10⁻⁷ (kz - t) i T . The wave passes through a cylindrical region of cross-sectional area 10 cm ^2 and length 50 cm along the z-axis. The total average energy contained in this cylindrical volume is: (Given ₀ = 4 10⁻⁷ T m A ⁻¹ )

Step-by-step solution

The total average energy density of an electromagnetic wave in terms of the magnetic field amplitude B₀ is given by: u_ avg = B₀^2 2 ₀ From the given equation, the amplitude of the magnetic field is B₀ = 8 10⁻⁷ T . Substituting the values: u_ avg = ( 8 10⁻⁷)^2 2 4 10⁻⁷ u_ avg = 8 10⁻¹⁴ 8 10⁻⁷ = 10⁻⁷ J m ⁻³ The volume of the cylindrical region is: V = Area Length V = (10 10⁻⁴ m ^2) (50 10⁻² m ) = 5 10⁻⁴ m ^3 The total average energy contained in the volume is: E = u_ avg V E = 10⁻⁷ 5 10⁻⁴ = 5 10⁻¹¹ J Answer: 5 10⁻¹¹