Question

The oscillating electric field in a plane electromagnetic wave is given by Comega%20t-kx%29%20V%2Fm%7D" id="TexFormula1" title="{50\sin (\omega t-kx) V/m}" alt="{50\sin (\omega t-kx) V/m}" align="absmiddle" class="latex-formula">, and the frequency of electric field is 2 × 10⁷ Hz. Find: (a) Find ${\omega}$, ${\lambda}$, k and write the expression for the electric field (b) Find ${B_{0}}$ and write the expression for magnetic field (c) Predict the Direction of propagation of electromagnetic wave.​

Answer 1

Here, we are given with:

${:\implies \quad \longrightarrow \begin{cases}\sf E_{0}= 50\: V/m\\ \sf \nu = 2\times 10^{7}Hz\end{cases}}$

(a) So now, we can thus obtain

${:\implies \quad \sf \omega =2\pi \nu}$

${:\implies \quad \sf \omega =2\pi (2\times 10^{7})}$

${:\implies \quad \boxed{\bf{\omega = 4\pi \times 10^{7}\: s^{-1}}}}$

Now, finding $\lambda$

${:\implies \quad \sf \lambda =\dfrac{c}{\omega}=\dfrac{3\times 10^{8}}{4\pi \times 10^{7}}}$

${:\implies \quad \boxed{\bf{\lambda \approx 2.39\:m}}}$

Now, finding k

${:\implies \quad \sf k=\dfrac{2\pi}{\lambda}=\dfrac{200\pi}{239}}$

${:\implies \quad \boxed{\bf{k\approx 2.63\:m^{-1}}}}$

Thus, expression for the electric field is:

${:\implies \quad \boxed{\bf{E=50\sin \bigg(4\pi \times 10^{7}t-\dfrac{263x}{100}\bigg)}}}$

(b) Now, here

${:\implies \quad \sf B_{0}=\dfrac{E_{0}}{c}=\dfrac{50}{3\times 10^{8}}}$

${:\implies \quad \boxed{\bf{B_{0}\approx 16.67×10^{-7}\:T}}}$

Thus, expression for the magnetic field:

${:\implies \quad \boxed{\bf{B_{y}=16.67\times 10^{-7}\sin \bigg(4\pi \times 10^{7}t-\dfrac{263x}{100}\bigg)\:T}}}$

(c) The electromagnetic wave propagates along Z-axis