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 wave, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Energy Densities**:
- The energy density (u) of the electric field (E) is given by the formula:
\[ u_E = \frac{1}{2} \epsilon_0 E_0^2 \]
- The energy density of the magnetic field (B) is given by:
\[ u_B = \frac{1}{2} \frac{B_0^2}{\mu_0} \]

2. **Equate the Energy Densities**:
- In an electromagnetic wave, the energy densities of the electric and magnetic fields are equal:
\[ u_E = u_B \]
- Therefore, we can set the two equations equal to each other:
\[ \frac{1}{2} \epsilon_0 E_0^2 = \frac{1}{2} \frac{B_0^2}{\mu_0} \]

3. **Cancel the Common Factors**:
- We can cancel the \(\frac{1}{2}\) from both sides:
\[ \epsilon_0 E_0^2 = \frac{B_0^2}{\mu_0} \]

4. **Rearrange the Equation**:
- Rearranging gives:
\[ E_0^2 = \frac{B_0^2}{\epsilon_0 \mu_0} \]

5. **Use the Speed of Light**:
- We know that the speed of light (c) is given by:
\[ c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} \]
- Therefore, we can write:
\[ \mu_0 \epsilon_0 = \frac{1}{c^2} \]

6. **Substitute Back**:
- Substituting this back into our equation gives:
\[ E_0^2 = B_0^2 c^2 \]

7. **Take the Square Root**:
- Taking the square root of both sides results in:
\[ E_0 = B_0 c \]

### Final Relationship:
Thus, the relationship between the amplitudes of the electric and magnetic fields in an electromagnetic wave is:
\[ E_0 = B_0 c \]