1. Identify the wave fields

A light wave is travelling along the $X$-axis and its electric field is $E_y = 900\sin\omega\left(t-\frac{x}{c}\right).$

So:

• Electric field is along $Y$-axis.
• Therefore magnetic field must be along $Z$-axis.
• For an electromagnetic wave, $B = \frac{E}{c}.$

2. Forces on the charge

A charge $q$ moving with velocity $\vec v$ experiences:

• Electric force: $F_E = qE$
• Magnetic force: $F_B = qvB\sin\theta$

Here the charge moves along the $Y$-axis, while magnetic field is along the $Z$-axis, so the angle between $\vec v$ and $\vec B$ is $90^\circ$. Thus, $F_B = qvB.$

3. Find the ratio $F_E : F_B$

Using $B = E/c$, $F_B = qv\frac{E}{c}.$

Hence, $\frac{F_E}{F_B} = \frac{qE}{qvE/c} = \frac{c}{v}.$

Given: $c = 3\times 10^8\,\text{m/s}, \qquad v = 3\times 10^7\,\text{m/s}.$

So, $\frac{F_E}{F_B} = \frac{3\times 10^8}{3\times 10^7} = 10.$

Therefore, $F_E : F_B = 10 : 1.$

4. Match with options

The correct option is:

C: $10:1$