The amplitudes of electric and magnetic fields are related to each other as

Options

A. `E_(0)B_(0)=omega`
B. `E_(0)= cB_(0)`
C. `E_(0)=(B_(0))/(c)`
D. `E_(0)=(c)/(B_(0))`

Answer

To determine the relationship between the amplitudes of the electric field (E₀) and the magnetic field (B₀) in an electromagnetic (EM) wave, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Maxwell's Equations**:
- The behavior of electromagnetic waves is governed by Maxwell's equations. One of the key equations is given by:
\[ \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \]
- This equation indicates that a changing electric field generates a magnetic field.

2. **Wave Equation Derivation**:
- From Maxwell's equations, we can derive the wave equations for electric and magnetic fields. For simplicity, we focus on the plane wave solutions.
- The wave equations can be expressed as:
\[ \frac{\partial^2 \mathbf{E}}{\partial t^2} = c^2 \nabla^2 \mathbf{E} \]
\[ \frac{\partial^2 \mathbf{B}}{\partial t^2} = c^2 \nabla^2 \mathbf{B} \]
- Here, \(c\) is the speed of light in a vacuum.

3. **Relating Amplitudes**:
- The general solutions for the electric and magnetic fields can be expressed as:
\[ \mathbf{E}(z, t) = E_0 \sin(kz - \omega t) \]
\[ \mathbf{B}(z, t) = B_0 \sin(kz - \omega t) \]
- Where \(E_0\) and \(B_0\) are the amplitudes of the electric and magnetic fields, respectively.

4. **Using the Wave Speed**:
- The speed of the wave \(c\) is related to the wave number \(k\) and angular frequency \(\omega\) by:
\[ c = \frac{\omega}{k} \]

5. **Finding the Relationship**:
- The relationship between the amplitudes can be derived from the equations:
\[ E_0 = c B_0 \]
- In terms of the speed of light in vacuum, this can be expressed as:
\[ E_0 = c B_0 \]
- Thus, we can express the relationship as:
\[ \frac{E_0}{B_0} = c \]

6. **Conclusion**:
- The final relationship between the amplitudes of the electric and magnetic fields in an electromagnetic wave is:
\[ E_0 = c B_0 \]
- This indicates that the amplitude of the electric field is equal to the speed of light multiplied by the amplitude of the magnetic field.