Question

In an electromagnetic wave in free space, the direction of electric field vector \(E^{-}\) is along \(y\) axis and magnetic field vector \(\mathrm{B}^{-}\) is along \(\mathrm{z}\) axis then which of the following is true (A) \(\left(\mathrm{E}^{-} \times \mathrm{B}^{-}\right) \times \mathrm{E}^{-}=1\) (B) \(\left(\mathrm{E}^{-} \times \mathrm{B}^{-}\right) \times \mathrm{B}^{-}=1\) (C) \(\left(\mathrm{E}^{-} \times \mathrm{B}^{-}\right) \times \mathrm{B}^{-}=0\) (D) none of these

Short answer

The correct answer is (C), which states that $\left(E^{-} \times B^{-}\right) \times B^{-} = 0$.

Step-by-step solution

Identify the given vectors

The electric field vector \(E^{-}\) is along the \(y\) axis, so it can be represented as: \[E^{-} = E_y \hat{j}\] The magnetic field vector \(B^{-}\) is along the \(z\) axis, so it can be represented as: \[B^{-} = B_z \hat{k}\]

Calculate the cross product of the given vectors

We need to calculate the cross product of the electric field vector and the magnetic field vector: \[(E^{-} \times B^{-}) = (E_y\hat{j}) \times (B_z\hat{k})\] To calculate the cross product of these vectors, we can use the following rule for cross product of unit vectors: \[\hat{i} \times \hat{j} = \hat{k}, \hat{j} \times \hat{k} = \hat{i}, \text{ and } \hat{k} \times \hat{i} = \hat{j}\] Using this rule, we find the cross product: \[(E^{-} \times B^{-}) = E_yB_z(\hat{j} \times \hat{k}) = E_yB_z\hat{i}\]

Evaluate the given options using the cross product

Now, let's evaluate the options one by one using the cross product of the given vectors: (A) \[\left(E^{-} \times B^{-}\right) \times E^{-}\] \[(E_yB_z\hat{i}) \times (E_y\hat{j})\] The above expression will not be equal to 1. (B) \[\left(E^{-} \times B^{-}\right) \times B^{-}\] \[(E_yB_z\hat{i}) \times (B_z\hat{k})\] The above expression will not be equal to 1. (C) \[\left(E^{-} \times B^{-}\right) \times B^{-}\] \[(E_yB_z\hat{i}) \times (B_z\hat{k})\] In this case, we have the cross product of two perpendicular unit vectors, which is equal to the third unit vector, so the expression becomes: \[E_yB_z^2\hat{j}\] Now, let's calculate the cross product with the magnetic field vector: \[E_yB_z^2(\hat{j} \times \hat{k}) = E_yB_z^2\hat{i}\] The above expression will be equal to 0. (D) none of these. Therefore, the correct answer is (C).

Key concepts

Electric Field

The electric field is a fundamental concept in physics, representing a region around a charged particle where a force would be exerted on other charges. It is a vector field, meaning it has both magnitude and direction. In the context of electromagnetic waves, the electric field can oscillate and create a wave pattern.

• Direction: In our exercise, the electric field vector, denoted as \(E^{-}\), is aligned along the \(y\)-axis.
• Representation: It is expressed as \(E^{-} = E_y \hat{j}\), where \(E_y\) is the magnitude and \(\hat{j}\) indicates the direction along the \(y\)-axis.

Understanding the direction and representation of the electric field is crucial for solving problems involving vector operations, such as the cross product with a magnetic field.

Magnetic Field

A magnetic field is another essential vector field in physics, representing the region where a magnetic force affects moving electric charges. Like the electric field, it is important in the study of electromagnetic waves.

• Direction: For the provided problem, the magnetic field vector \(B^{-}\) is found along the \(z\)-axis.
• Representation: It is represented by \(B^{-} = B_z \hat{k}\), where \(B_z\) is the magnitude and \(\hat{k}\) signifies the direction along the \(z\)-axis.

The interaction between electric and magnetic fields is fundamental in generating and propagating electromagnetic waves, demonstrating how changes in one field can induce changes in the other.

Cross Product

The cross product is a vector operation used in mathematics and physics to find a vector that is perpendicular to two other vectors. It is significant in the study of electromagnetic waves where electric and magnetic fields interact.

• Formula: The cross product of two vectors \(\mathbf{A}\) and \(\mathbf{B}\) is given as \(\mathbf{A} \times \mathbf{B}\), resulting in a new vector.
• Direction and Magnitude: The resultant vector is perpendicular to the plane formed by the original vectors and its magnitude is \(|\mathbf{A}| |\mathbf{B}| \sin(\theta)\), where \(\theta\) is the angle between \(\mathbf{A}\) and \(\mathbf{B}\).
• Order and Properties: The cross product is anti-commutative, meaning \(\mathbf{A} \times \mathbf{B} = -\mathbf{B} \times \mathbf{A}\). This property is useful for determining directional relationships.

In the exercise, we utilized the cross product to determine the orientation and interactions of the electric and magnetic fields, further evaluating cross products of cross products.

Vector Calculus

Vector calculus is a branch of mathematics focused on vector fields and operations, including differentiation and integration. It provides tools for understanding and applying concepts related to electric and magnetic fields.

• Operations: Important operations include the gradient, divergence, and curl, which help describe physical phenomena.
• Applications: In electromagnetic theory, vector calculus is employed to analyze and quantify fields' behaviors and interactions.
• Importance: For electromagnetic waves, vector calculus helps to solve complex field interactions and equations, leading to a profound understanding of wave propagation.

Particularly, evaluating the cross product ~\(\mathbf{E}^{-} \times \mathbf{B}^{-}\)~ relies on knowledge of vector calculus to analyze the relationship and find solutions to problems like the given exercise.