Suppose that the electric field amplitude of an electromagnetic wave us `E_0=120 N// C` and that its frequency is `50.0 MHz`.
(a) Determine` B_0,omega,k` and `lambda`,
(b) find expressions for `E` and `B`.

Explanation

To solve the given problem step by step, we will determine the magnetic field amplitude \( B_0 \), angular frequency \( \omega \), wave number \( k \), wavelength \( \lambda \), and then find the expressions for the electric field \( E \) and magnetic field \( B \).

### Step-by-Step Solution:

**Given:**
- Electric field amplitude \( E_0 = 120 \, \text{N/C} \)
- Frequency \( f = 50.0 \, \text{MHz} = 50.0 \times 10^6 \, \text{Hz} \)

### Part (a): Determine \( B_0, \omega, k, \lambda \)

1. **Calculate \( B_0 \)**:
The relationship between the electric field amplitude \( E_0 \) and the magnetic field amplitude \( B_0 \) in an electromagnetic wave is given by:
\[ B_0 = \frac{E_0}{c} \]
where \( c \) is the speed of light (\( c \approx 3 \times 10^8 \, \text{m/s} \)).
\[ B_0 = \frac{120 \, \text{N/C}}{3 \times 10^8 \, \text{m/s}} = 4 \times 10^{-7} \, \text{T} \]

2. **Calculate \( \omega \)**:
The angular frequency \( \omega \) is related to the frequency \( f \) by:
\[ \omega = 2\pi f \]
Substituting the value of \( f \):
\[ \omega = 2\pi \times 50.0 \times 10^6 \approx 314159.27 \, \text{rad/s} \]

3. **Calculate \( k \)**:
The wave number \( k \) is given by:
\[ k = \frac{\omega}{c} \]
Substituting the values of \( \omega \) and \( c \):
\[ k = \frac{314159.27}{3 \times 10^8} \approx 1.047 \times 10^{-3} \, \text{m}^{-1} \]

4. **Calculate \( \lambda \)**:
The wavelength \( \lambda \) can be calculated using the relationship:
\[ \lambda = \frac{c}{f} \]
Substituting the values:
\[ \lambda = \frac{3 \times 10^8}{50.0 \times 10^6} = 6 \, \text{m} \]

### Summary of Part (a):
- \( B_0 = 4 \times 10^{-7} \, \text{T} \)
- \( \omega \approx 314159.27 \, \text{rad/s} \)
- \( k \approx 1.047 \times 10^{-3} \, \text{m}^{-1} \)
- \( \lambda = 6 \, \text{m} \)

### Part (b): Find expressions for \( E \) and \( B \)

1. **Expression for \( E \)**:
The general expression for the electric field in an electromagnetic wave is:
\[ E(x, t) = E_0 \sin(\omega t - kx) \]
Substituting the known values:
\[ E(x, t) = 120 \sin(314159.27 t - 1.047 x) \]

2. **Expression for \( B \)**:
The general expression for the magnetic field is:
\[ B(x, t) = B_0 \sin(\omega t - kx) \]
Substituting the known values:
\[ B(x, t) = 4 \times 10^{-7} \sin(314159.27 t - 1.047 x) \]

### Final Expressions:
- \( E(x, t) = 120 \sin(314159.27 t - 1.047 x) \)
- \( B(x, t) = 4 \times 10^{-7} \sin(314159.27 t - 1.047 x) \)