Magnetic field in a region through which electromagnetic waves are propagating is given by : B=Bocos(kz)cos(wt)j cap Average value of poynting vector is?

Question

Answer

In order to find the average value of the Poynting vector for the given magnetic field of an electromagnetic wave, we will follow these steps:

Understanding the Magnetic Field
- The given magnetic field is:
**B = B₀ cos(kz) cos(ωt) ĵ**
- Here,
- **B₀** is the amplitude of the magnetic field,
- **k** is the wave number,
- **ω** is the angular frequency,
- **z** is the propagation direction,
- **t** represents time,
- **ĵ** indicates the direction of the magnetic field.

Finding the Electric Field
- The electric field **E** in an electromagnetic wave can be related to the magnetic field using:
**E = cB**
where **c** is the speed of light.
- Thus, the electric field can be expressed as:
**E = E₀ cos(kz) cos(ωt) î**
where **E₀ = cB₀**.

Calculating the Poynting Vector
- The Poynting vector **S** is given by:
**S = (1/μ₀) (E × B)**
- Substituting the electric and magnetic fields, we find:
**S = (1/μ₀) (E₀ cos(kz) cos(ωt) î × B₀ cos(kz) cos(ωt) ĵ)**
- This results in:
**S = (E₀B₀/μ₀) cos²(kz) cos²(ωt) k̂**

Average Value of Poynting Vector
- The average value of **S** over one complete cycle can be calculated as:
**⟨S⟩ = (E₀B₀/μ₀) (1/2)(1/2) = (E₀B₀)/(4μ₀)**
- Using **E₀ = cB₀**, we can further simplify:
**⟨S⟩ = (B₀²)/(4μ₀c)**
This average value represents the energy flux of the electromagnetic wave propagating in the z-direction.

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