Question

What are electromagnetic waves? Give a brief description of the main properties of electromagnetic waves. The equation for oscillation of the magnetic field of an electromagnetic wave is \( B_y = 8 \times 10^{-6} \sin \left( 2 \times 10^{11} t + 300 \pi x \right)~\text{T \). Find the wavelength and equation for the oscillating electric field. Mention the direction of propagation of the wave also.}

Hint:

In electromagnetic waves, the direction of propagation is perpendicular to both the electric and magnetic field directions. The relationship between the magnetic field and electric field is \( E_0 = c B_0 \), where \( c \) is the speed of light.

Answer 1

Electromagnetic waves are a combination of electric and magnetic fields that oscillate perpendicular to each other and to the direction of propagation. They travel at the speed of light, \( c = 3 \times 10^8~\text{m/s} \), and do not require a medium for propagation.
The general equation for a magnetic field in an electromagnetic wave is given by:
\[ B_y = B_0 \sin(kx - \omega t) \] Where:
- \(B_y\) is the magnetic field,
- \(B_0\) is the maximum magnetic field (amplitude),
- \(k\) is the wave number,
- \(\omega\) is the angular frequency,
- \(x\) is the position along the x-axis, and
- \(t\) is time.
From the given equation \( B_y = 8 \times 10^{-6} \sin \left( 2 \times 10^{11} t + 300 \pi x \right)~\text{T} \), we can compare the terms with the standard wave equation. Here:
- \(B_0 = 8 \times 10^{-6}~\text{T}\),
- The angular frequency \( \omega = 2 \times 10^{11}~\text{rad/s} \),
- The wave number \( k = 300 \pi~\text{rad/m} \).
Now, the wavelength \( \lambda \) is related to the wave number \( k \) by the equation:
\[ k = \frac{2\pi}{\lambda} \] Substitute \( k = 300 \pi \):
\[ 300 \pi = \frac{2\pi}{\lambda} $\Rightarrow$ \lambda = \frac{2}{300} = 6.67 \times 10^{-3}~\text{m} \] Next, the electric field \( E \) oscillates in the same direction as the magnetic field but is perpendicular to it. The magnitude of the electric field is related to the magnetic field by the equation:
\[ E_0 = c B_0 \] Substitute \( c = 3 \times 10^8~\text{m/s} \) and \( B_0 = 8 \times 10^{-6}~\text{T} \):
\[ E_0 = (3 \times 10^8) \times (8 \times 10^{-6}) = 2400~\text{V/m} \] Thus, the equation for the oscillating electric field is:
\[ E_y = E_0 \sin \left( 2 \times 10^{11} t + 300 \pi x \right) \] Where \( E_0 = 2400~\text{V/m} \).
The direction of propagation is along the \( x \)-axis, which is determined by the cross-product of the electric field and magnetic field vectors. Thus, the wave is traveling in the positive \( x \)-direction.