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A sinusoidal electromagnetic wave is propagating in vacuum in the +\(z\)-direction. If at a particular instant and at a certain point in space the electric field is in the +\(x\)-direction and has magnitude 4.00 V/m, what are the magnitude and direction of the magnetic field of the wave at this same point in space and instant in time?

Short Answer

Expert verified
The magnetic field is 1.33 x 10^-8 T in the +y-direction.

Step by step solution

01

Understand the Relationship Between Electric and Magnetic Fields in an Electromagnetic Wave

In a vacuum, the electric field (\(E\)) and the magnetic field (\(B\)) of an electromagnetic wave are perpendicular to each other and to the direction of wave propagation. The magnitude of the magnetic field can be found using the relationship: \[ B = \frac{E}{c} \]where \(c\) is the speed of light in a vacuum, approximately \(3 \times 10^8\) m/s.
02

Substitute the Given Values into the Formula

We know the magnitude of the electric field \(E = 4.00\) V/m. We will substitute \(E\) and \(c\) into the formula:\[ B = \frac{4.00}{3 \times 10^8} \]
03

Calculate the Magnitude of the Magnetic Field

Perform the division to find \(B\): \[ B = \frac{4.00}{3 \times 10^8} = 1.33 \times 10^{-8} \text{ T} \]
04

Determine the Direction of the Magnetic Field

The direction of the magnetic field (\(B\)) is perpendicular to both the electric field (\(E\)) and the direction of wave propagation. Given that \(E\) points in the +\(x\)-direction and the wave propagates in the +\(z\)-direction, use the right-hand rule to determine that \(B\) is directed in the +\(y\)-direction.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Electric Field
The electric field is a fundamental concept in electromagnetism. Imagine it as lines of force that emanate from charged particles and interact with other charged particles nearby. In the context of electromagnetic waves, these fields are oscillating back and forth. These changes propagate through space and can influence other electrically charged particles.

Key features of electric fields include:
  • Direction: The electric field is a vector quantity, meaning it has both magnitude and direction.
  • Structure: It often appears as a sinusoidal wave when described in the context of electromagnetic radiation.
  • Interaction: The field can cause other charges to feel a force, pushing or pulling them along the direction of the field.
In our exercise, the electric field is in the +\(x\)-direction with a magnitude of 4.00 V/m. This means at that point in space, a positive test charge would experience a force pushing it in the positive x-direction.
Magnetic Field
The magnetic field is another vector field, closely related to the electric field in electromagnetic waves. It forms a right-angle with the electric field and the direction of wave propagation, acting as a companion to the electric field.

It's crucial to know these about magnetic fields:
  • Perpendicularity: In an electromagnetic wave, the magnetic field is always perpendicular to the electric field.
  • Magnitude: The magnitude of the magnetic field \(B\) can be derived from the electric field \(E\) by the relationship \( B = \frac{E}{c} \), where \(c\) is the speed of light.
  • Induced motion: It affects moving charges, inducing a force orthogonal to their motion.
For the given problem, we calculated the magnitude of the magnetic field to be \(1.33 \times 10^{-8}\) T, and determined its direction using the right-hand rule.
Right-Hand Rule
The right-hand rule is a simple yet powerful mnemonic for finding the orientation of magnetic fields in relation to electric fields. This rule helps in determining directions in a three-dimensional space which is common when dealing with electromagnetic waves.

To use the right-hand rule in our exercise:
  • Point your thumb in the direction of the wave's propagation (the +\(z\)-direction).
  • Extend your index finger in the direction of the electric field (the +\(x\)-direction).
  • Your middle finger, when extended at a right angle to these two, indicates the direction of the magnetic field (the +\(y\)-direction).
This rule ensures that you keep binary relationships of perpendicularity consistent, letting you visualize where each field vector points in an electromagnetic wave, helping solve complex problems with ease.

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Most popular questions from this chapter

A small helium-neon laser emits red visible light with a power of 5.80 mW in a beam of diameter 2.50 mm. (a) What are the amplitudes of the electric and magnetic fields of this light? (b) What are the average energy densities associated with the electric field and with the magnetic field? (c) What is the total energy contained in a 1.00-m length of the beam?

In the 25-ft Space Simulator facility at NASA's Jet Propulsion Laboratory, a bank of overhead arc lamps can produce light of intensity 2500 W/m\(^2\) at the floor of the facility. (This simulates the intensity of sunlight near the planet Venus.) Find the average radiation pressure (in pascals and in atmospheres) on (a) a totally absorbing section of the floor and (b) a totally reflecting section of the floor. (c) Find the average momentum density (momentum per unit volume) in the light at the floor.

In a certain experiment, a radio transmitter emits sinusoidal electromagnetic waves of frequency 110.0 MHz in opposite directions inside a narrow cavity with reflectors at both ends, causing a standing-wave pattern to occur. (a) How far apart are the nodal planes of the magnetic field? (b) If the standing- wave pattern is determined to be in its eighth harmonic, how long is the cavity?

Electromagnetic waves propagate much differently in conductors than they do in dielectrics or in vacuum. If the resistivity of the conductor is sufficiently low (that is, if it is a sufficiently good conductor), the oscillating electric field of the wave gives rise to an oscillating conduction current that is much larger than the displacement current. In this case, the wave equation for an electric field \(\vec{E} (x, t) = E_y(x, t)\hat{\jmath}\) en propagating in the +\(x\)-direction within a conductor is $${\partial^2E_y(x, t)\over \partial x^2} = {\mu \over \rho} {\partial Ey(x, t)\over \partial t}$$ where \(\mu\) is the permeability of the conductor and \(\rho\) is its resistivity. (a) A solution to this wave equation is \(E_y(x, t) = E_{max} e^{-k_C x} cos(k_Cx - \omega t)\), where \(k_C = \sqrt{(\omega \mu/2\rho}\). Verify this by substituting E_y(x, t) into the above wave equation. (b) The exponential term shows that the electric field decreases in amplitude as it propagates. Explain why this happens. (\(Hint\): The field does work to move charges within the conductor. The current of these moving charges causes \(i^2R\) heating within the conductor, raising its temperature. Where does the energy to do this come from?) (c) Show that the electric-field amplitude decreases by a factor of 1/\(e\) in a distance \(1/k_C = \sqrt{2\rho/\omega\mu}\), and calculate this distance for a radio wave with frequency \(f\) = 1.0 MHz in copper (resistivity 1.72 \(\times\) 10\(^{-8 } \Omega \bullet m\); permeability \(\mu = \mu_0\)). Since this distance is so short, electromagnetic waves of this frequency can hardly propagate at all into copper. Instead, they are reflected at the surface of the metal. This is why radio waves cannot penetrate through copper or other metals, and why radio reception is poor inside a metal structure.

An electromagnetic wave has an electric field given by \(\vec{E} (y, t)\) = (3.10 \(\times\) 10\(^5\) V/m) \(\hat{k}\) cos [ky - (12.65 \(\times\) 10\(^{12}\) rad/s)t]. (a) In which direction is the wave traveling? (b) What is the wavelength of the wave? (c) Write the vector equation for \(\vec{B} (y, t)\).

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