Question

An electromagnetic wave going through vacuum is described by $E=E_{0} \sin (k x-\omega t)\( then \)B=B_{0} \sin (k x-\omega t)$ then (A) \(E_{0} B_{0}=\operatorname{cok}\) (B) \(E_{0} k=B_{0} \omega\) (C) \(\mathrm{E}_{0} \mathrm{~m}=\mathrm{B}_{0} \mathrm{k}\) (D) none of these

Short answer

The correct answer is (D) none of these because the relation we derived between the amplitudes of electric and magnetic fields (\(E_0 / B_0 = c\)) does not match any of the given options.

Step-by-step solution

Understand the values of variables in the given equations

In the given equations, we have \(E = E_0 \sin(kx - \omega t)\) and \(B = B_0 \sin(kx - \omega t)\). Here, \(E_0\) and \(B_0\) are the amplitudes of the electric and magnetic fields, respectively; k is the wave vector, \(\omega\) is the angular frequency, x is the position, and t is the time.

Compare the coefficients of sin function in both given equations

Now, compare the coefficients of the sin function in both E and B: \(E=E_0 \sin(kx-\omega t) \Rightarrow E_0 = E /\sin(kx - \omega t)\) \(B=B_0 \sin(kx-\omega t) \Rightarrow B_0 = B /\sin(kx - \omega t)\)

Use the relation between the magnitudes of electric and magnetic fields and check the given options

We know the relation between magnitudes of electric and magnetic fields: \(E = cB\). From Step 2, we can write this relation in terms of the amplitudes: \(E_0 = cB_0 \Rightarrow E_0 / B_0 = c\) Now let's check the options for the relation between \(E_0\) and \(B_0\): (A) \(E_0 B_0 = cok\) ⟹ This option is not consistent with our derived relation. (B) \(E_0 k = B_0 \omega\) ⟹ From our derived relation, if k = c, then this option is correct. However, we cannot assume this, so we cannot determine whether this option is correct or not. (C) \(E_0 m = B_0 k\) ⟹ This option is not consistent with our derived relation. (D) None of these ⟹ Since our derived relation is not matching with any provided options, this option is the correct answer. Hence, the correct answer is (D) none of these.