Final answer:
The directivity of an antenna measures how well it focuses the radiated power in a particular direction. To find the directivity of the given antenna with the E-field pattern independent of ϕ, we need to calculate the total radiation power in the maximum direction and the total radiation power that would be radiated if the antenna had uniform radiation in all directions. The directivity is then given by the ratio of these two powers.
Step-by-step explanation:
The directivity of an antenna measures how well it focuses the radiated power in a particular direction. It is typically expressed as the ratio of the power radiated in the maximum direction to the power that would be radiated if the antenna had the same total power output but radiated it equally in all directions.
To find the directivity of the given antenna with the E-field pattern independent of ϕ, we need to calculate the total radiation power in the maximum direction and the total radiation power that would be radiated if the antenna had uniform radiation in all directions. Then, we can divide the former by the latter to get the directivity.
Since the E-field pattern given is symmetric about the y-axis, the maximum direction is in the y-axis direction (perpendicular to the axis of the antenna). The power radiated in the maximum direction is half the total power output, since the E-field has a value of 1/2 in the range 0° ≤ θ ≤ 90°. The power radiated if the antenna had uniform radiation in all directions is the average power over the entire sphere surrounding the antenna, which can be calculated by integrating the square of the E-field magnitude over the entire range of θ and ϕ. By evaluating this integral, we can find the total radiation power. The directivity is then given by the ratio of these two powers.