Final answer:
The induced electric field vector within a solenoid can be determined using Faraday's law of electromagnetic induction. The amplitude of the time-varying current through the solenoid can be calculated using the equation I(t) = I0cos(ωt). By plotting both amplitudes as a function of time, we can observe the relationship between the two quantities.
Step-by-step explanation:
The induced electric field vector as a function of radius within a solenoid can be determined by using Faraday's law of electromagnetic induction. The induced electric field is given by the equation E = -dΦ/dt, where Φ is the magnetic flux through a surface bounded by the path of integration. In the case of a solenoid, the magnetic field is constant within the solenoid, so the magnitude of the induced electric field is constant within the solenoid as well. The direction of the induced electric field depends on the direction of the change in magnetic field.
To calculate the amplitude of the time-varying current through the solenoid, we can use the equation I(t) = I0cos(ωt), where I0 is the maximum amplitude of the current and ω is the angular frequency. By plotting the amplitude of the time-varying current and the amplitude of the induced electric field as a function of time, we can observe the relationship between the two quantities.