Question

A dipole antenna is located at the origin with its axis along the \(z\) -axis. As electric current oscillates up and down the antenna, polarized electromagnetic radiation travels away from the antenna along the positive \(y\) -axis. What are the possible directions of electric and magnetic fields at point \(A\) on the \(y\) -axis? Explain.

Short answer

Answer: The possible directions of the electric and magnetic fields at point A on the y-axis are as follows: 1. Electric field (E-field): \(E_x\hat{x}\) and Magnetic field (B-field): \(B_z\hat{z}\) 2. Electric field (E-field): \(E_z\hat{z}\) and Magnetic field (B-field): \(B_x\hat{x}\)

Step-by-step solution

Understanding the given information

We have a dipole antenna located at the origin, with its axis along the z-axis. The polarized electromagnetic radiation travels away from the antenna along the positive y-axis. We need to determine the possible directions for the electric and magnetic fields at any point A on the y-axis.

Recognizing the relationship between electric and magnetic fields

In electromagnetic radiation, the electric field (E-field) and magnetic field (B-field) are always perpendicular (orthogonal) to each other and to the direction of propagation. This means that if the radiation is propagating along the y-axis, both the electric and magnetic fields must be in the plane formed by the x-axis and z-axis.

Determine the possible directions of the electric field (E-field)

Since we know that the E-field must be orthogonal to the direction of propagation (y-axis) and confined to the plane formed by the x-axis and z-axis, the possible directions of the E-field can only be along the x-axis or the z-axis. This can be represented as either \(\vec{E} = E_x\hat{x}\) or \(\vec{E} = E_z\hat{z}\).

Determine the possible directions of the magnetic field (B-field)

As mentioned earlier, the B-field must also be orthogonal to the direction of propagation (y-axis) and confined to the plane formed by the x-axis and z-axis. Since the B-field must also be orthogonal to the E-field, this means that if the E-field is along the x-axis, the B-field must be along the z-axis, and vice versa. The possible directions of the B-field can be represented as either \(\vec{B} = B_x\hat{x}\) or \(\vec{B} = B_z\hat{z}\).

Present the possible directions of E-field and B-field

The possible directions of electric and magnetic fields at any point A on the y-axis are determined as follows: 1. \(\vec{E} = E_x\hat{x}\) and \(\vec{B} = B_z\hat{z}\) 2. \(\vec{E} = E_z\hat{z}\) and \(\vec{B} = B_x\hat{x}\) These combinations ensure that the electric and magnetic fields are orthogonal to each other and to the direction of propagation along the y-axis.