Question

Equation of electric field of a plane electromagnetic wave is given as:
\[ E_z = 60 \sin(500x + 1.5 \times 10^{11} t) \, \text{V/m}. \] Write the equation for the magnetic field of the wave.

Hint:

In electromagnetic waves, the electric and magnetic fields are always perpendicular to each other and to the direction of wave propagation. The relationship between the two fields is given by \( \frac{E_z}{B_z} = c \), where \( c \) is the speed of light in a vacuum. The magnetic field can be found by dividing the electric field by \( c \).

Answer 1

To find the magnetic field of the electromagnetic wave, we need to utilize the relationship between the electric and magnetic fields in a plane electromagnetic wave. In such a wave, the electric field \( \vec{E} \) and the magnetic field \( \vec{B} \) are related by the speed of light \( c \), and they are perpendicular to each other, as well as to the direction of propagation of the wave.
For an electromagnetic wave traveling along the \( x \)-axis, the electric and magnetic fields are related by the following equation:
\[ \frac{E_z}{B_z} = c \] Where:
- \( E_z \) is the electric field,
- \( B_z \) is the magnetic field,
- \( c \) is the speed of light in a vacuum, which is \( 3 \times 10^8 \, \text{m/s} \).
Thus, the magnetic field can be written as:
\[ B_z = \frac{E_z}{c} \] Given that the electric field is: \[ E_z = 60 \sin(500x + 1.5 \times 10^{11} t) \, \text{V/m}, \] we can substitute this into the equation for \( B_z \):
\[ B_z = \frac{60 \sin(500x + 1.5 \times 10^{11} t)}{3 \times 10^8} \, \text{T}. \] Now, simplifying the expression:
\[ B_z = \frac{60}{3 \times 10^8} \sin(500x + 1.5 \times 10^{11} t) \, \text{T}. \] We can calculate the constant factor:
\[ \frac{60}{3 \times 10^8} = 2 \times 10^{-7}. \] Thus, the equation for the magnetic field becomes:
\[ B_z = 2 \times 10^{-7} \sin(500x + 1.5 \times 10^{11} t) \, \text{T}. \] Interpretation of the Result:
- The magnetic field \( B_z \) has the same functional form as the electric field \( E_z \), since they are both sinusoidal waves with the same frequency and wavevector.
- The only difference is the amplitude, which is scaled by a factor of \( 2 \times 10^{-7} \), due to the relationship between the electric and magnetic fields in an electromagnetic wave.
- The phase of the wave, as well as the wavevector \( k \) and angular frequency \( \omega \), remain unchanged.
- The magnetic field oscillates in sync with the electric field but is scaled by the factor related to the speed of light.