Final answer:
To find the rate at which the current in the solenoid must change to create the specified electric field, one can use the magnetic field equation for a solenoid and apply Faraday's law, using the known parameters of the solenoid and the radial distance.
Step-by-step explanation:
To determine the rate at which the current must change in the solenoid to create a desired electric field at a particular radial distance, one must use the concept of the magnetic field produced by a solenoid and Maxwell's equations, specifically the one that correlates a changing magnetic field with an induced electric field (Faraday's law of electromagnetic induction). The equation for the magnetic field inside an infinitely long solenoid is B = μ_0 * n * I, where μ_0 is the permeability of free space, n is the number of turns per unit length, and I is the current. In this case, the electric field E at a radial distance r from the axis of the solenoid is related to the rate of change of the magnetic field within the solenoid.
To solve the problem, we would need to apply Faraday's law that relates the induced electric field around a closed loop to the rate of change of the magnetic flux through the loop. The magnitude of this electric field E is given by the formula E = r / 2 * (d/dt), where r is the radial distance from the axis of the solenoid to the point of interest and d/dt is the rate of change of the magnetic field. Therefore, the rate of change of the current dI/dt can be calculated by rearranging the formula to solve for dI/dt using the provided electric field and the known parameters of the solenoid (number of windings per meter and the radial distance).