At one instant, the electric and magnetic fields at one point of an electromagnetic wave are E=(200i^+300j^−50k^) V/m\mathbf{E} = (200 \hat{\mathbf{i}} + 300 \hat{\mathbf{j}} - 50 \hat{\mathbf{k}}) \text{ V/m} and B=B0(7.3i^−7.3j^+ak^ μT\mathbf{B}=$$B_0(7.3$$\hat{\mathbf{i}}-7.3\hat{\mathbf{j}}+a\hat{\mathbf{k}}\text{ }\mu T. What is the Poynting vector at this time and position?

Ch 31: Electromagnetic Fields and Waves

Knight Calc - Physics for Scientists and Engineers 5th Edition

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Knight Calc 5th Edition

Ch 31: Electromagnetic Fields and Waves

Problem 41b

Chapter 31, Problem 41b

At one instant, the electric and magnetic fields at one point of an electromagnetic wave are $E = (200 \hat{i} + 300 \hat{j} - 50 \hat{k}) V/m$ and $B = B_{0} (7.3 \hat{i} - 7.3 \hat{j} + a \hat{k} μ T$ . What is the Poynting vector at this time and position?

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The Poynting vector $ \mathbf{S} $ represents the power per unit area carried by an electromagnetic wave and is given by $ \mathbf{S} = \frac{1}{\mu_0} \mathbf{E} \times \mathbf{B} $, where $ \mu_0 $ is the permeability of free space, $ \mathbf{E} $ is the electric field, and $ \mathbf{B} $ is the magnetic field.

Substitute the given electric field $ \mathbf{E} = (200 \hat{\mathbf{i}} + 300 \hat{\mathbf{j}} - 50 \hat{\mathbf{k}}) \text{ V/m} $ and the magnetic field $ \mathbf{B} = B_0(7.3 \hat{\mathbf{i}} - 7.3 \hat{\mathbf{j}} + a \hat{\mathbf{k}}) \text{ } \mu T $ into the cross product formula $ \mathbf{E} \times \mathbf{B} $.

Perform the cross product $ \mathbf{E} \times \mathbf{B} $ using the determinant method: $ \mathbf{E} \times \mathbf{B} = \begin{vmatrix} \hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \\ 200 & 300 & -50 \\ 7.3B_0 & -7.3B_0 & aB_0 \end{vmatrix} $. Expand this determinant to compute the components of the resulting vector.

Simplify the determinant to find the components of $ \mathbf{E} \times \mathbf{B} $. Remember to carefully handle the terms involving $ B_0 $ and $ a $. The result will be a vector in terms of $ B_0 $ and $ a $.

Finally, divide the resulting vector by $ \mu_0 $ to compute the Poynting vector $ \mathbf{S} $. Note that $ \mu_0 = 4\pi \times 10^{-7} \text{ T·m/A} $. The final expression for $ \mathbf{S} $ will depend on $ B_0 $ and $ a $.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Electromagnetic Fields

Electromagnetic fields consist of electric fields (E) and magnetic fields (B) that propagate through space. The electric field is represented as a vector quantity, indicating the force per unit charge, while the magnetic field represents the force on moving charges. In the context of electromagnetic waves, these fields oscillate perpendicular to each other and to the direction of wave propagation.

Poynting Vector

The Poynting vector (S) quantifies the directional energy flux of an electromagnetic field, defined as S = E × B/μ₀, where E is the electric field, B is the magnetic field, and μ₀ is the permeability of free space. It indicates the power per unit area carried by the wave and points in the direction of energy flow. Understanding the Poynting vector is essential for analyzing the energy transfer in electromagnetic waves.

Vector Cross Product

The vector cross product is a mathematical operation that takes two vectors and produces a third vector that is perpendicular to the plane formed by the original vectors. In the context of the Poynting vector, the cross product of the electric field vector and the magnetic field vector determines the direction and magnitude of energy flow in an electromagnetic wave. This operation is crucial for calculating the Poynting vector accurately.

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At one instant, the electric and magnetic fields at one point of an electromagnetic wave are $\vec{E} = (200 \hat{i} + 300 \hat{j} - 50 \hat{k}) V/m$ and $\vec{B} = B_{0} (7.3 \hat{i} - 7.3 \hat{j} + a \hat{k}) μ T$ . What are the values of $a$ and $B sub 0$ ?

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