Question

The electric field associated with an electromagnetic radiation is given by E = a(1 + cosω₁t)cosω₂t. Which of the following frequencies are present in the field?

• A. ω₁
• B. ω₁+ω₂
• C. |ω₁+ω₂|
• D. ω₂

Answer 1

The Correct Option is B, C, D

To determine the frequencies present in the given electric field, let's start by analyzing the expression provided:

The electric field is given by: \(E = a(1 + \cos \omega_1 t) \cos \omega_2 t\)

We need to expand this expression using trigonometric identities. Specifically, we'll use the product-to-sum identities for simplifying the product of cosine terms. The relevant identity here is:

\(\cos A \cos B = \frac{1}{2}[\cos(A + B) + \cos(A - B)]\)

Applying this identity, we get:

\(E = a[\cos \omega_2 t + \cos \omega_1 t \cos \omega_2 t]\)

Breaking down further, we expand the term \(\cos \omega_1 t \cos \omega_2 t\) using the identity:

\(\cos \omega_1 t \cos \omega_2 t = \frac{1}{2}[\cos(\omega_1 + \omega_2)t + \cos(\omega_1 - \omega_2)t]\)

Substituting back, we have:

\(E = a[\cos \omega_2 t + \frac{1}{2}(\cos(\omega_1 + \omega_2)t + \cos(\omega_1 - \omega_2)t)]\)

This expression shows that the frequencies present in the electric field are:

• \(\omega_2\)
• \(\omega_1 + \omega_2\)
• \(\omega_1 - \omega_2\)

Thus, the options that correctly list the frequencies present are:

• \(\omega_2\)
• \(\omega_1 + \omega_2\)
• \(|\omega_1 + \omega_2|\)

Therefore, the correct answer is:

• \(\omega_1 + \omega_2\)
• \(|\omega_1 + \omega_2|\)
• \(\omega_2\)