The question asks us to find the frequency at which the conduction current in the medium is equal to the displacement current. To solve this, we need to understand the concepts of conduction current and displacement current.

1. Conduction Current:
Conduction current is the flow of electric charge through a conductor, usually in response to an applied electric field. It is governed by Ohm's law, which states that the current flowing through a conductor is directly proportional to the applied electric field and inversely proportional to the resistance of the conductor. The formula for conduction current is given by Ic = σE, where Ic is the conduction current, σ is the conductivity of the material, and E is the electric field.

2. Displacement Current:
Displacement current is a concept introduced by James Clerk Maxwell to explain the behavior of electric fields in the absence of conduction currents. It arises from time-varying electric fields and is related to the rate of change of electric flux through a surface. The formula for displacement current is given by Id = ε₀(dE/dt), where Id is the displacement current, ε₀ is the permittivity of free space, and dE/dt is the rate of change of electric field.

Now, let's solve the problem step by step:

Given data:
Conductivity (σ) = 10^-2 mho/m
Relative permittivity (εᵣ) = 4

Step 1: Calculate the permittivity of the material
The permittivity (ε) of the material is given by ε = ε₀εᵣ, where ε₀ is the permittivity of free space (8.854 x 10^-12 F/m). Therefore, ε = (8.854 x 10^-12 F/m) * 4 = 3.5416 x 10^-11 F/m.

Step 2: Equate the conduction current and displacement current
To find the frequency at which the conduction current is equal to the displacement current, we need to equate the two currents. Therefore, σE = ε₀(dE/dt).

Step 3: Simplify the equation
Since σ = 10^-2 mho/m and ε₀ = 8.854 x 10^-12 F/m, we can substitute these values into the equation. The equation becomes (10^-2 mho/m) * E = (8.854 x 10^-12 F/m) * (dE/dt).

Step 4: Solve for frequency
We know that frequency (f) is given by f = ω/2π, where ω is the angular frequency. The angular frequency is related to the rate of change of the electric field (dE/dt) by the equation ω = 2πf. Therefore, we can rewrite the equation as (10^-2 mho/m) * E = (8.854 x 10^-12 F/m) * ω.

Step 5: Substitute values and solve for frequency
Substituting the given values into the equation, we get (10^-2 mho/m) * E = (8.854 x 10^-12 F/m) * 2πf. Rearranging the equation, we get f = (10^-2 mho/m) * E / [(8.854 x 10^-12 F/m) *