The electric field in a radio wave traveling through vacuum has amplitude $2.5 \times 10^{-4} \mathrm{V} / \mathrm{m}\( and frequency \)1.47 \mathrm{MHz}$. (a) Find the amplitude and frequency of the magnetic field. (b) The wave is traveling in the \(+x\) -direction. At \(x=0\) and \(t=0,\) the electric field is \(1.5 \times 10^{-4} \mathrm{V} / \mathrm{m}\) in the \- y-direction. What are the magnitude and direction of the magnetic field at \(x=0\) and \(t=0 ?\).

Using the relationship between the amplitudes of electric and magnetic fields in an electromagnetic wave, we have: \(B_0 = \frac{E_0}{c}\) Where \(E_0\) is the amplitude of the electric field (2.5 x 10^(-4) V/m), c is the speed of light in a vacuum (3 x 10^8 m/s), and \(B_0\) is the amplitude of the magnetic field. \(B_0 = \frac{2.5 \times 10^{-4}}{3.0 \times 10^8} = 8.33 \times 10^{-13} \mathrm{T}\) So the amplitude of the magnetic field is 8.33 x 10^(-13) T.

Since the frequency of the electric field and the magnetic field are the same in an electromagnetic wave, the frequency of the magnetic field is: \(f_B = 1.47 \mathrm{MHz}\)

To find the magnitude and direction of the magnetic field at x=0 and t=0, we first need to find the phase of the electric field. Given that the electric field is in the y-direction, its expression is: \(E_y = E_0 \sin(2 \pi f_t- kx + \phi)\) Where \(E_y = 1.5 \times 10^{-4} \mathrm{V/m}\) when x=0 and t=0. Since the electric field is in the y-direction, we can say that the magnetic field is in the z-direction. Therefore, we can write the magnetic field expression as: \(B_z = B_0 \sin(2 \pi f_t - kx +\phi)\) Plugging in the values given: \(E_y = 2.5 \times 10^{-4} \sin(\phi)\) \(1.5 \times 10^{-4} = 2.5 \times 10^{-4} \sin(\phi)\) Solving for φ, we get: \(\phi = \arcsin{\frac{1.5 \times 10^{-4}}{2.5 \times 10^{-4}}} \approx 0.6435 \, rad\) Now, we can find the magnetic field at x=0 and t=0 by plugging in the values for the magnetic field expression: \(B_z = B_0 \sin(\phi) = (8.33 \times 10^{-13} \mathrm{T}) \sin(0.6435) \approx 6.75 \times 10^{-13} \mathrm{T}\) Thus, the magnetic field's magnitude at x=0 and t=0 is approximately 6.75 x 10^(-13) T, and its direction is along the +z-axis. In summary: a) The amplitude of the magnetic field is 8.33 x 10^(-13) T, and the frequency is 1.47 MHz. b) The magnitude of the magnetic field at x=0 and t=0 is approximately 6.75 x 10^(-13) T, and its direction is along the +z-axis.