Electric field of electromagnetic wave is E =800sin(π( 10⁸t + x/150 ))V/m where x is in cm and t in seconds. A charged particle is moving with speed 1.5×10⁶m/s. Find the ratio of magnetic force to electric force on the charge particle.

The correct option is 10^-2.

Step-by-step Explanation:

We are given the equation for the electric field of an electromagnetic wave:

$E=800sin\left(\pi \left({10}^{8}t+\frac{x}{150}\right)\right)\text{ V/m}$

where $t$ is in seconds and $x$ is in centimeters (cm).

To analyze the wave propagation, we first convert the position coordinate $x$ from centimeters to meters. Since $x_{\text{cm}}=100x_{\text{m}}$, we substitute this relationship into the phase argument of the sine wave:

$\frac{x_{\text{cm}}}{150}=\frac{100x_{\text{m}}}{150}=\frac{x_{\text{m}}}{1.5}$

Thus, the equation in SI units (where $x$ is in meters) becomes:

$E=800sin\left(\pi {10}^{8}t+\frac{\pi }{1.5}x\right)\text{ V/m}$

Comparing this with the standard wave equation form $E=E_{0}sin\left(\omega t+kx\right)$, we identify:
Angular frequency:

$\omega =\pi \times {10}^8\text{ rad/s}$

Wave number:

$k=\frac{\pi }{1.5}\text{ rad/m}$

The speed of propagation of the electromagnetic wave ($c$) is given by:

$c=\frac{\omega }{k}=\frac{\pi \times {10}^8}{\frac{\pi }{1.5}}$

$c=1.5\times {10}^8\text{ m/s}$

For an electromagnetic wave, the relation between the electric field intensity ($E$) and magnetic field intensity ($B$) at any instant is:

$B=\frac{E}{c}$

Next, let a particle with charge $q$ move with a speed $v=1.5\times {10}^6\text{ m/s}$ through this wave field.
The electric force acting on the charge is:

$F_e=qE$

The maximum magnetic force acting on the moving charge is:

$F_m=qvB$

We are asked to find the ratio of the magnetic force to the electric force:

$\frac{F_m}{F_e}=\frac{qvB}{qE}=v\left(\frac{B}{E}\right)$

Since $\frac{B}{E}=\frac{1}{c}$, the ratio simplifies to:

$\frac{F_m}{F_e}=\frac{v}{c}$

Substituting the values of $v$ and $c$:

$\frac{F_m}{F_e}=\frac{1.5\times {10}^6\text{ m/s}}{1.5\times {10}^8\text{ m/s}}={10}^{-2}$