∙Predict/Calculate A light beam traveling in the negative z...

$\bullet$ Predict/Calculate $A$ light beam traveling in the negative $z$ direction has a magnetic field $\overline{\mathbf{B}}=\left(3.02 \times 10^{-9} \mathrm{T}\right) \hat{\mathbf{x}}+$ $\left(-5.28 \times 10^{-9} \mathrm{T}\right) \hat{\mathbf{y}}$ at a given instant of time. The electric field in the beam has a magnitude of 1.82 $\mathrm{N} / \mathrm{C}$ at the same time. (a) Does the electric field at this time have a $z$ component that is positive, negative, or zero? Explain. (b) Write $\overline{\mathbf{E}}$ in terms of unit vectors.

$\bullet$ Predict/Calculate $A$ light beam traveling in the negative $z$ direction has a magnetic field $\overline{\mathbf{B}}=\left(3.02 \times 10^{-9} \mathrm{T}\right) \hat{\mathbf{x}}+$
$\left(-5.28 \times 10^{-9} \mathrm{T}\right) \hat{\mathbf{y}}$ at a given instant of time. The electric field in the beam has a magnitude of 1.82 $\mathrm{N} / \mathrm{C}$ at the same time. (a) Does
the electric field at this time have a $z$ component that is positive,
negative, or zero? Explain. (b) Write $\overline{\mathbf{E}}$ in terms of unit vectors.

Key Concepts

Electromagnetic Wave Propagation

Electromagnetic waves are disturbances that propagate through space, carrying energy in the form of oscillating electric and magnetic fields. These fields, as described by Maxwell's equations, do not have components in the direction of the wave's travel, and instead oscillate in a plane perpendicular to the propagation direction.

Transverse Nature of Fields

In light beams and other electromagnetic waves, the electric and magnetic fields are transverse. This means that if the wave propagates along one axis (for instance, the z-axis), then both the electric and magnetic fields must lie entirely in the perpendicular plane (the xy plane). This inherent characteristic ensures that there is no field component along the propagation direction.

Right-Hand Rule in Electromagnetism

The right-hand rule is a fundamental concept used to determine the orientation of the electric field, magnetic field, and the direction of wave propagation in an electromagnetic wave. Specifically, if you point the fingers of your right hand in the direction of the electric field and curl them toward the direction of the magnetic field, your thumb will point in the direction of wave propagation.

Transcript

00:01 Okay, so in this problem we have a magnetic vector field with components of 3 .02 times 10 to the minus 9 x hat minus let's see 5 .28 5 .28 times 10 to the minus 9 y hat.

00:36 This is the magnetic vector field.

00:39 Let's rewrite in here.

00:40 It's a little odd this b.

00:45 Okay, this is better.

00:48 What else we know? we also know that the electric field has a magnitude.

00:59 Let's put in here.

01:02 Electric field has a magnitude of 1 .82 newton's per column, should be a c.

01:14 And the first question of this problem we just need to answer what is the z component of the electric field okay so again same as the problem 85 let's write this down coordinate system with the z direction in here a y direction in here and x direction in here.

01:46 If we know that this wave has a propagation in the positive z direction and let's think about that the magnetic field is only in the y direction.

02:04 We know that the electric field should be only in the x direction because the two are always perpendicular to each other.

02:14 So if the the b is in the xy plan, the electric field should also be in the xy plan.

02:26 And we know that the z component, so let's put in here the first item, the z hat component of the electric field is zero because he does not have any type of components in the z direction.

02:51 So now let's calculate what should be the values of the components of the electric field...

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