In a plane electromagnetic wave, the electric field oscillates sinusoidally with an amplitude of $48 ext{ V/m}$ and a frequency of $5.0 imes 10^{14} ext{ Hz}$. (a) Calculate the amplitude of the oscillating magnetic field ($B0$). (b) Calculate the wavelength of the wave ($\lambda$). (c) Write the expressions for the electric field vector and magnetic field vector if the wave is propagating along the positive z-axis.

Q: In a plane electromagnetic wave, the electric field oscillates sinusoidally with an amplitude of $48\text{ V/m}$ and a frequency of $5.0 \times 10^{14}\text{ Hz}$. (a) Calculate the amplitude of the oscillating magnetic field ($B0$). (b) Calculate the wavelength of the wave ($\lambda$). (c) Write the expressions for the electric field vector and magnetic field vector if the wave is propagating along the positive z-axis.

Given Data: Amplitude of electric field, $E0 = 48\text{ V/m}$ Frequency, $\nu = 5.0 \times 10^{14}\text{ Hz}$ Speed of light in vacuum, $c = 3 \times 10^8\text{ m/s}$ --- Part (a): Calculate the amplitude of the magnetic field ($B0$) \nThe relation between the amplitude of the electric field and the magnetic field in an electromagnetic wave is given by: $$c = \frac{E0}{B0}$$ \nRearranging the formula to find $B0$: $$B0 = \frac{E0}{c}$| \nSubstituting the given values: $$B0 = \frac{48\text{ V/m}}{3 \times 10^8\text{ m/s}}$$ $$B0 = 1.6 \times 10^{-7}\text{ T}$$ --- Part (b): Calculate the wavelength of the wave ($\lambda$) \nThe wave equation connecting speed, frequency, and wavelength is: $$c = \nu \lambda$$ \nRearranging for wavelength $\lambda$: $$\lambda = \frac{c}{\nu}$$ \nSubstituting the given values: $$\lambda = \frac{3 \times 10^8\text{ m/s}}{5.0 \times 10^{14}\text{ Hz}}$$ $$\lambda = 0.6 \times 10^{-6}\text{ m} = 600\text{ nm}$$ --- Part (c): Write the expressions for electric and magnetic fields \nLet us assume the electric field vector oscillates along the x-axis, i.e., $\vec{E} = Ex \hat{i}$.\nThen, the magnetic field must oscillate along the y-axis, i.e., $\vec{B} = By \hat{j}$, since the wave propagates along the z-axis ($\hat{k} = \hat{i} \times \hat{j}$). \nAngular frequency $\omega = 2\pi \nu$: $$\omega = 2 \times \pi \times 5.0 \times 10^{14} = 3.14 \times 10^{15}\text{ rad/s}$| \nPropagation constant $k = \frac{2\pi}{\lambda}$: $$k = \frac{2 \times \pi}{600 \times 10^{-9}} = 1.05 \times 10^7\text{ rad/m}$| Electric Field Expression: $$\vec{E}(z, t) = 48 \sin(1.05 \times 10^7 z - 3.14 \times 10^{15} t)\hat{i}\text{ V/m}$| Magnetic Field Expression: $$\vec{B}(z, t) = 1.6 \times 10^{-7} \sin(1.05 \times 10^7 z - 3.14 \times 10^{15} t)\hat{j}\text{ T}$$

Question

In a plane electromagnetic wave, the electric field oscillates sinusoidally with an amplitude of $48	ext{ V/m}$ and a frequency of $5.0 	imes 10^{14}	ext{ Hz}$. (a) Calculate the amplitude of the oscillating magnetic field ($B0$). (b) Calculate the wavelength of the wave ($\lambda$). (c) Write the expressions for the electric field vector and magnetic field vector if the wave is propagating along the positive z-axis.

Accepted Answer

Given Data: Amplitude of electric field, $E0 = 48	ext{ V/m}$ Frequency, $
u = 5.0 	imes 10^{14}	ext{ Hz}$ Speed of light in vacuum, $c = 3 	imes 10^8	ext{ m/s}$ --- Part (a): Calculate the amplitude of the magnetic field ($B0$) 
The relation between the amplitude of the electric field and the magnetic field in an electromagnetic wave is given by: $$c = \frac{E0}{B0}$$ 
Rearranging the formula to find $B0$: $$B0 = \frac{E0}{c}$| 
Substituting the given values: $$B0 = \frac{48	ext{ V/m}}{3 	imes 10^8	ext{ m/s}}$$ $$B0 = 1.6 	imes 10^{-7}	ext{ T}$$ --- Part (b): Calculate the wavelength of the wave ($\lambda$) 
The wave equation connecting speed, frequency, and wavelength is: $$c = 
u \lambda$$ 
Rearranging for wavelength $\lambda$: $$\lambda = \frac{c}{
u}$$ 
Substituting the given values: $$\lambda = \frac{3 	imes 10^8	ext{ m/s}}{5.0 	imes 10^{14}	ext{ Hz}}$$ $$\lambda = 0.6 	imes 10^{-6}	ext{ m} = 600	ext{ nm}$$ --- Part (c): Write the expressions for electric and magnetic fields 
Let us assume the electric field vector oscillates along the x-axis, i.e., $\vec{E} = Ex \hat{i}$.
Then, the magnetic field must oscillate along the y-axis, i.e., $\vec{B} = By \hat{j}$, since the wave propagates along the z-axis ($\hat{k} = \hat{i} 	imes \hat{j}$). 
Angular frequency $\omega = 2\pi 
u$: $$\omega = 2 	imes \pi 	imes 5.0 	imes 10^{14} = 3.14 	imes 10^{15}	ext{ rad/s}$| 
Propagation constant $k = \frac{2\pi}{\lambda}$: $$k = \frac{2 	imes \pi}{600 	imes 10^{-9}} = 1.05 	imes 10^7	ext{ rad/m}$| Electric Field Expression: $$\vec{E}(z, t) = 48 \sin(1.05 	imes 10^7 z - 3.14 	imes 10^{15} t)\hat{i}	ext{ V/m}$| Magnetic Field Expression: $$\vec{B}(z, t) = 1.6 	imes 10^{-7} \sin(1.05 	imes 10^7 z - 3.14 	imes 10^{15} t)\hat{j}	ext{ T}$$

Given Data:

Part (a): Calculate the amplitude of the magnetic field ($B_0$)

Part (b): Calculate the wavelength of the wave ($\lambda$)

Part (c): Write the expressions for electric and magnetic fields