At one instant, the electric and magnetic fields at one point of an electromagnetic wave are E→=(200i^+300j^−50k^) V/m\overrightarrow{E}=(200\hat{i}+300\hat{j}-50\hat{k})\text{ V/m} and B→=B0(7.3i^−7.3j^+ak^) μT\overrightarrow{B}=$$B_0(7.3$$\hat{i}-7.3\hat{j}+a\hat{k})\text{ }\mu\text{T}. What are the values of aa and B0B_0?

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 41a

Chapter 31, Problem 41a

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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The electric field vector $ \overrightarrow{E} $ and the magnetic field vector $ \overrightarrow{B} $ in an electromagnetic wave are always perpendicular to each other and to the direction of wave propagation. This means $ \overrightarrow{E} \cdot \overrightarrow{B} = 0 $ (dot product is zero for perpendicular vectors). Use this condition to find the relationship between $ a $ and $ B_0 $.

Write the dot product of $ \overrightarrow{E} $ and $ \overrightarrow{B} $: $ \overrightarrow{E} \cdot \overrightarrow{B} = (200)(7.3B_0) + (300)(-7.3B_0) + (-50)(aB_0) = 0 $. Simplify this equation to solve for $ a $.

The magnitudes of $ \overrightarrow{E} $ and $ \overrightarrow{B} $ are related by the speed of light $ c $: $ \frac{|\overrightarrow{E}|}{|\overrightarrow{B}|} = c $. Calculate $ |\overrightarrow{E}| $ using $ |\overrightarrow{E}| = \sqrt{200^2 + 300^2 + (-50)^2} $.

Express $ |\overrightarrow{B}| $ in terms of $ B_0 $: $ |\overrightarrow{B}| = B_0 \sqrt{(7.3)^2 + (-7.3)^2 + a^2} $. Use the relationship $ \frac{|\overrightarrow{E}|}{|\overrightarrow{B}|} = c $ to solve for $ B_0 $.

Combine the results from the dot product condition and the magnitude relationship to find the values of $ a $ and $ B_0 $.

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Key Concepts

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

Electromagnetic Waves

Electromagnetic waves are oscillations of electric and magnetic fields that propagate through space. They are characterized by their frequency, wavelength, and speed, which is the speed of light in a vacuum. The electric field (E) and magnetic field (B) are perpendicular to each other and to the direction of wave propagation, following Maxwell's equations.

Maxwell's Equations

Maxwell's equations describe how electric and magnetic fields interact and propagate. They consist of four fundamental equations that relate electric charges, currents, and the fields they produce. These equations predict that a changing electric field generates a magnetic field and vice versa, which is the basis for the propagation of electromagnetic waves.

Vector Representation of Fields

Electric and magnetic fields are represented as vectors, indicating both magnitude and direction. In the given problem, the electric field vector E and the magnetic field vector B are expressed in terms of their components along the Cartesian coordinates (i, j, k). Understanding vector addition and the relationships between these components is crucial for solving for unknowns like B0 and a.

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Textbook Question

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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Textbook Question

When the Voyager 2 spacecraft passed Neptune in 1989, it was 4.5×10⁹ km from the earth. Its radio transmitter, with which it sent back data and s, broadcast with a mere 21 W of power. Assuming that the transmitter broadcast equally in all directions, What signal intensity was received on the earth?

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Textbook Question

A 1.0 μF capacitor is discharged, starting at t = 0 s.The displacement current between the plates is $I_{disp} = (10 A) \exp (- \frac{t}{2.0} μ s)$ . What was the capacitor’s initial voltage (ΔV_C)₀?

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Textbook Question

What is the total energy density in an electromagnetic wave of intensity 1000 W/m²?

Textbook Question

A 10 A current is charging a 1.0-cm-diameter parallel-plate capacitor. What is the magnetic field strength at a point 2.0 mm radially from the center of the capacitor?

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Textbook Question

FIGURE P31.38 shows the electric field inside a cylinder of radius $R equals 3.0$ mm. The field strength is increasing with time as $E = 1.0 \times 10^{8} t^{2}$ V/m, where t is in s. The electric field outside the cylinder is always zero, and the field inside the cylinder was zero for $t is less than 0$ . Find an expression for the electric flux $Φ_{e}$ through the entire cylinder as a function of time.

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