Question

The magnetic field in a microwave traveling through vacuum has amplitude $4.00 \times 10^{-11} \mathrm{T}\( and frequency \)120 \mathrm{GHz}$. Find the amplitude and frequency of the electric field.

Short answer

Question: Given the amplitude and frequency of the magnetic field of a microwave traveling through a vacuum, calculate the amplitude and frequency of the electric field. Answer: The amplitude of the electric field is 1.20 x 10^-2 N/C and its frequency is 120 x 10^9 Hz.

Step-by-step solution

Recall the relationship between the electric field and the magnetic field in an electromagnetic wave

In a vacuum, the relationship between the electric field amplitude (\(E_0\)) and the magnetic field amplitude (\(B_0\)) is given by the equation: $$E_0 = cB_0$$ where \(c\) is the speed of light in a vacuum, approximately equal to \(3.00 \times 10^8\ \text{m/s}\).

Calculate the electric field amplitude

Given the magnetic field amplitude \(B_0 = 4.00 \times 10^{-11}\ \mathrm{T}\), we can find the electric field amplitude using the equation from Step 1: $$E_0 = cB_0$$ $$E_0 = (3.00 \times 10^8\ \mathrm{m/s})(4.00 \times 10^{-11}\ \mathrm{T})$$ $$E_0 = 1.20 \times 10^{-2}\ \mathrm{N/C}$$ Therefore, the amplitude of the electric field in the microwave is \(1.20 \times 10^{-2}\ \mathrm{N/C}\).

Identify the frequency of the electric field

Since the frequency of the electric field is the same as the frequency of the magnetic field, we can directly say that the frequency of the electric field is \(120\ \mathrm{GHz}\). It's important to note that, in the context of electromagnetic waves, frequency usually should be expressed in Hz. Converting GHz to Hz, we have: $$120\ \mathrm{GHz} = 120 \times 10^9\ \mathrm{Hz}$$ Thus, the frequency of the electric field is \(120 \times 10^9\ \mathrm{Hz}\). In summary, the amplitude of the electric field is \(1.20 \times 10^{-2}\ \mathrm{N/C}\) and its frequency is \(120 \times 10^9\ \mathrm{Hz}\).