Problem 2141 - JEE Physics

Question

What is the direction of \(\mathrm{E}^{-} \times \mathrm{B}^{-}\) in an electromagnetic wave? (A) same as that of \(E^{-}\) (B) same as that of \(\mathrm{B}^{-}\) (C) same as the direction of propagation of electromagnetic wave (D) none of these

Step-by-Step Solution

Answer

The direction of \(\mathrm{E}^{-} \times \mathrm{B}^{-}\) in an electromagnetic wave is the same as the direction of propagation of the electromagnetic wave. Therefore, the correct answer is (C).

Step 1: Recall the right-hand rule for cross products

The right-hand rule states that if you hold your right hand with your thumb, index, and middle fingers mutually perpendicular and let your thumb point in the direction of the first vector and your index finger point in the direction of the second vector, then your middle finger will point in the direction of the cross product.

Step 2: Remember the properties of electromagnetic waves

In an electromagnetic wave, the electric field, \(E^{-}\), and the magnetic field, \(B^{-}\), are always perpendicular to each other and also to the direction of propagation of the wave.

Step 3: Apply the right-hand rule to find the direction of \(\mathrm{E}^{-} \times \mathrm{B}^{-}\)

Use the right-hand rule as described in Step 1. Point your thumb in the direction of the electric field, \(E^{-}\), and your index finger in the direction of the magnetic field, \(B^{-}\). Your middle finger, representing the cross product, will point in the direction of the propagation of the electromagnetic wave.

Step 4: Choose the correct answer

Based on the results from Step 3, the direction of \(\mathrm{E}^{-} \times \mathrm{B}^{-}\) is the same as the direction of the propagation of the electromagnetic wave. Thus, the correct answer is (C) same as the direction of propagation of electromagnetic wave.

Key Concepts

Right-Hand Rule

The right-hand rule is a helpful technique used to determine the direction of vectors resulting from cross products, such as those often found in physics, particularly electromagnetic wave equations. It's quite simple:

• Hold your right hand out in front of you with your fingers spread.
• Point your thumb in the direction of the first vector (in this case, the electric field, \(E^{-}\)).
• Then point your index finger in the direction of the second vector (the magnetic field, \(B^{-}\)).
• Your middle finger, now perpendicular to both your thumb and index finger, will point in the direction of the third vector, which is the cross product \(E^{-} \times B^{-}\).

This rule is very effective for visualizing how the directions of vectors interplay in three-dimensional space.

Cross Product

The 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. Here’s what you need to know:

• It's denoted as \(\mathbf{A} \times \mathbf{B}\).
• This product follows the right-hand rule for direction.
• Its magnitude is given by \(|\mathbf{A}| \times |\mathbf{B}| \times \sin(\theta)\), where \(\theta\) is the angle between \(\mathbf{A}\) and \(\mathbf{B}\).
• This is particularly useful in physics, for determining the resultant vector of forces like in electromagnetic waves where the fields are perpendicular to each other.

Electric Field Direction

In electromagnetic waves, the electric field, denoted as \(E^{-}\), is a vector quantity that describes the electric force experienced by a positive charge. It’s essential to grasp the direction for these fields:

• Electric fields point in the direction that a positive test charge would move.
• In electromagnetic waves, \(E^{-}\) is perpendicular to both the magnetic field \(B^{-}\) and the direction of wave propagation.
• For visualization, this means if you consider the wave moving forward, \(E^{-}\) shoots out sideways.

Magnetic Field Direction

The magnetic field, represented by \(B^{-}\), is another vector quantity inherent to electromagnetic waves. Understanding its direction involves several simple points:

• Magnetic fields exert forces on moving charges or other magnetic fields.
• In an electromagnetic wave, the magnetic field is perpendicular to both the electric field \(E^{-}\) and the direction of the wave’s propagation.
• This implies that if you see the wave traveling forward, \(B^{-}\) extends out perpendicularly in a different plane from \(E^{-}\).

Wave Propagation Direction

The direction of wave propagation in electromagnetic waves is a crucial point and can be found using the cross product \(E^{-} \times B^{-}\). Here’s how it works:

• In electromagnetic waves, this direction is perpendicular to both the electric and magnetic fields.
• Using the right-hand rule, with your thumb pointing in the direction of \(E^{-}\) and your index in the \(B^{-}\) direction, your middle finger will show the wave propagation direction.
• This means the electromagnetic wave moves forward in the direction of the resultant vector \(E^{-} \times B^{-}\).

Understanding this relationship is fundamental as it ties together both the electric and magnetic components to define wave movement.