To analytically calculate the electric field at a distance \( r \) from a time-varying current-carrying wire, we can utilize Maxwell's equations, particularly focusing on the relationship between the changing magnetic field and the induced electric field. Let's break this down step by step.

Understanding the Magnetic Field from a Current-Carrying Wire

When a current \( I(t) \) flows through a wire, it generates a magnetic field around it. For a straight wire, the magnetic field \( B \) at a distance \( r \) from the wire can be expressed using the Biot-Savart law. In this case, the magnetic field is given by:

B = (μ₀/4π) * (I(t) * sin(θ₁) + I(t) * sin(θ₂))

Here, \( μ₀ \) is the permeability of free space, and \( θ₁ \) and \( θ₂ \) are the angles corresponding to the ends of the wire. The direction of the magnetic field follows the right-hand rule, where your thumb points in the direction of the current and your fingers curl in the direction of the magnetic field lines.

Calculating the Electric Field Induced by a Changing Magnetic Field

According to Faraday's law of electromagnetic induction, a changing magnetic field induces an electric field. Mathematically, this is represented as:

curl E = -dB/dt

To find the electric field \( E \) at a distance \( r \), we first need to compute the time derivative of the magnetic field \( B \). Since the current \( I(t) \) is time-varying, we can express \( B \) as a function of time:

B(t) = (μ₀/4π) * I(t) * (sin(θ₁) + sin(θ₂))

Now, differentiating \( B(t) \) with respect to time gives us:

dB/dt = (μ₀/4π) * (dI(t)/dt) * (sin(θ₁) + sin(θ₂))

Applying the Curl to Find the Electric Field

Substituting \( dB/dt \) back into Faraday's law, we have:

curl E = - (μ₀/4π) * (dI(t)/dt) * (sin(θ₁) + sin(θ₂))

To find the electric field \( E \), we can use the cylindrical symmetry of the problem. For a long straight wire, the electric field will be azimuthal (in the \( \phi \) direction) and can be expressed in terms of the distance \( r \) from the wire:

E = (1/(2πr)) * ∫(curl E) · dA

Given the symmetry, we can simplify this to:

E = - (μ₀/4π) * (dI(t)/dt) * (1/r) * (sin(θ₁) + sin(θ₂))

Final Expression for the Electric Field

Thus, the electric field \( E \) at a distance \( r \) from the wire, induced by the time-varying current, can be expressed as:

E(r, t) = - (μ₀/4π) * (dI(t)/dt) * (1/r) * (sin(θ₁) + sin(θ₂))

This equation shows how the electric field depends on the rate of change of the current \( I(t) \), the distance \( r \) from the wire, and the angles \( θ₁ \) and \( θ₂ \) associated with the geometry of the wire. The negative sign indicates the direction of the induced electric field is opposite to the change in magnetic field, in accordance with Lenz's law.

In summary, by applying Faraday's law and considering the geometry of the wire, we can analytically derive the electric field induced by a time-varying current-carrying wire. This relationship is fundamental in understanding electromagnetic induction and the interplay between electric and magnetic fields.