Question

If $E_o$ and $B_o$ are the magnitudes of the electric and magnetic fields respectively of an electromagnetic wave in vacuum, then among the following the correct relation is ($\mu_o$ -- permeability of free spa, $\varepsilon_o$ -- permittivity of free spa)

• A. $E_o = B_o \sqrt{\mu_o \varepsilon_o}$
• B. $E_o \varepsilon_o = B_o \mu_o$
• C. $E_o \sqrt{\varepsilon_o} = \frac{B_o}{\sqrt{\mu_o}}$
• D. $\frac{E_o}{\sqrt{\varepsilon_o}} = \frac{B_o}{\sqrt{\mu_o}}$

Hint:

Electromagnetic Wave Properties:

• $E_0 = cB_0$
• $c = \frac1\sqrt\mu_0 \varepsilon_0$
• Therefore, $E_0 = \fracB_0\sqrt\mu_0 \varepsilon_0$
• Rearranged: $E_0 \sqrt\varepsilon_0 = \fracB_0\sqrt\mu_0$

Answer 1

The Correct Option is C

For an electromagnetic wave in vacuum, the relation between $E_0$ and $B_0$ is: \[ E_0 = c B_0 \quad \text{and} \quad c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}} \] Substitute $c$ into the first equation: \[ E_0 = \frac{B_0}{\sqrt{\mu_0 \varepsilon_0}} \Rightarrow E_0 \sqrt{\varepsilon_0} = \frac{B_0}{\sqrt{\mu_0}} \] Thus, the correct relationship is given in option (3).

Answer 2

Step 1: Understand the relation between electric and magnetic fields in an electromagnetic wave
In a vacuum, the electric field \( E_o \) and magnetic field \( B_o \) of an electromagnetic wave are related through the speed of light \( c \).

Step 2: Recall the speed of light formula in terms of permittivity and permeability
The speed of light in vacuum is given by:
\[ c = \frac{1}{\sqrt{\mu_o \varepsilon_o}} \]

Step 3: Relation between electric and magnetic fields
For an electromagnetic wave in vacuum:
\[ E_o = c B_o \]

Step 4: Substitute the value of \( c \)
Using the expression of \( c \), we get:
\[ E_o = \frac{B_o}{\sqrt{\mu_o \varepsilon_o}} \]

Step 5: Rearranging the equation
Multiply both sides by \( \sqrt{\varepsilon_o} \):
\[ E_o \sqrt{\varepsilon_o} = \frac{B_o}{\sqrt{\mu_o}} \]

Step 6: Conclusion
Hence, the correct relation between \( E_o \) and \( B_o \) is:
\[ E_o \sqrt{\varepsilon_o} = \frac{B_o}{\sqrt{\mu_o}} \]