Determine EM Wave Frequency Using Wave Zeros
An electromagnetic (EM) wave traveling in vacuum moves at the speed of light, $c = 3 \times 10^8$ m/s. The electric field of a monochromatic plane polarized wave propagating along the z-direction can be expressed as:
$ E(z, t) = E_0 \sin(kz - \omega t + \phi) $
Here, $k$ is the wave number ($k = \frac{2\pi}{\lambda}$) and $\omega$ is the angular frequency ($\omega = 2\pi f$). The wave's electric field is zero when the argument of the sine function equals an integer multiple of $\pi$:
$ kz - \omega t + \phi = n\pi $
where $n$ is an integer.
Relating Spatial Zeros to Wavelength
We are given that at a specific time $t_1$, the electric field is zero at a spatial point $z_1$. This condition can be written as:
$ kz_1 - \omega t_1 + \phi = n_1\pi \quad (1) $
The problem states that the *next* zero in its neighborhood occurs at $z_2$. Assuming $z_2$ is the next spatial point where the field is zero at the same time $t_1$, the corresponding phase must be the next integer multiple of $\pi$. Therefore, $n_2 = n_1 + 1$:
$ kz_2 - \omega t_1 + \phi = (n_1 + 1)\pi \quad (2) $
Subtracting equation (1) from equation (2) eliminates $\omega t_1$ and $\phi$, giving the relationship between the spatial separation and the wave number:
$ (kz_2 - \omega t_1 + \phi) - (kz_1 - \omega t_1 + \phi) = (n_1 + 1)\pi - n_1\pi $
$ k(z_2 - z_1) = \pi $
Calculating the Frequency
Substitute the wave number $k = \frac{2\pi}{\lambda}$ into the derived equation:
$ \left(\frac{2\pi}{\lambda}\right)(z_2 - z_1) = \pi $
This equation simplifies to:
$ \frac{2}{\lambda}(z_2 - z_1) = 1 $
From this, we find the wavelength $\lambda$ in terms of the spatial separation:
$ \lambda = 2(z_2 - z_1) $
The frequency $f$ of an electromagnetic wave is related to its speed $c$ and wavelength $\lambda$ by the equation $c = f\lambda$. Rearranging to solve for frequency:
$ f = \frac{c}{\lambda} $
Substitute the value of $c$ for vacuum ($c = 3 \times 10^8$ m/s) and the derived expression for $\lambda$:
$ f = \frac{3 \times 10^8 \text{ m/s}}{2(z_2 - z_1)} $
$ f = \frac{1.5 \times 10^8}{z_2 - z_1} $
The frequency of the electromagnetic wave is $\frac{1.5\times10^8}{z_2-z_1}$.
Similar Questions
- A short bar magnet placed with its axis at $30^\circ$ with an external field of 800 Gauss, experiences a torque of 0.016 N.m. The work done in moving it from most stable to most unstable position is $\alpha \times 10^{-3} \text{ J}$. The value of $\alpha$ is ______.
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Important Questions from Electricity and Magnetism
- A short bar magnet placed with its axis at $30^\circ$ with an external field of 800 Gauss, experiences a torque of 0.016 N.m. The work done in moving it from most stable to most unstable position is $\alpha \times 10^{-3} \text{ J}$. The value of $\alpha$ is ______.
- An electromagnetic wave of frequency $100\text{ MHz}$ propagates through a medium of conductivity, $\sigma = 10\text{ mho/m}$. The ratio of maximum conduction current density to maximum displacement current density is __________.
$\left[ \text{Take } \frac{1}{4\pi\varepsilon_0} = 9 \times 10^9\text{ Nm}^2\text{/C}^2 \right]$ - An inductor stores $16\text{ J}$ of magnetic field energy and dissipates $32\text{ W}$ of thermal energy due to its resistance when an a.c. current of $2\text{ A}$ (rms) and frequency $50\text{ Hz}$ flows through it. The ratio of inductive reactance to its resistance is _________. ($\pi = 3.14$)
- A simple pendulum made of mass 10 g and a metallic wire of length 10 cm is suspended vertically in a uniform magnetic field of 2 T. The magnetic field direction is perpendicular to the plane of oscillations of the pendulum. If the pendulum is released from an angle of $60^\circ$ with vertical, then maximum induced EMF between the point of suspension and point of oscillation is ___________ mV. (Take $g = 10\text{ m/s}^2$)
Using a variable frequency a.c. voltage source the maximum current measured in the given LCR circuit is 50 mA for $V = 5\sin(100t)$ The values of L and R are shown in the figure. The capacitance of the capacitor (C) used is ___________ $\mu\text{F}$.
