Question

What is the corresponding magnetic field h(r,θ,t) for the given electric field e(r,θ,t)=θ²×10−2rsinθcos(6π×108t−2πr) (v/m)?

a) h(r, θ, t) = -r² × 10⁻²r cos(θ) cos(6π × 10⁸t - 2πr) (A/m)

b) h(r, θ, t) = -ϕ² × 10⁻²r sin(θ) sin(6π × 10⁸t - 2πr) (A/m)

c) h(r, θ, t) = -z² × 10⁻²r cos(θ) sin(6π × 10⁸t - 2πr) (A/m)

d) h(r, θ, t) = θ² × 10⁻²r sin(θ) sin(6π × 10⁸t - 2πr) (A/m)

Answer 1

The corresponding magnetic field to a given electric field in an electromagnetic wave must consider Maxwell's equations and the fact that electric and magnetic fields are orthogonal and in phase. The options presented in the question may be incorrect or incomplete, and a confident determination of the magnetic field requires further context and the application of these physical principles.

Step-by-step explanation:

The question involves finding the magnetic field corresponding to a given electric field. The relationship between electric field and magnetic field in electromagnetic waves is dictated by Maxwell's equations. Specifically, the magnetic field h(r, θ, t) can be determined by the curl of the electric field, or in simpler terms, by considering that electromagnetic waves have orthogonal electric and magnetic fields which are in phase.

In this particular case, we are dealing with fields in cylindrical coordinates and a time-varying scenario that suggests a form of a traveling wave.

Given that the question asks for a relation and no explicit derivation is provided, one can only attempt to find the correct option based on the concept that the magnetic field components must be orthogonal to the electric field and proportionate in magnitude, considering wave propagation direction and medium impedance.

Out of the options provided, none of them consists of fields exactly orthogonal to each other and in the correct propagation direction implied by the initial electric field. Without further context or supporting derivation, we cannot confidently determine the corresponding magnetic field.

Options provided might be incorrect or incomplete, and an accurate determination of the magnetic field requires use of appropriate Maxwell's equations and understanding of electromagnetic wave propagation.