Given data:
- Diameter of the wire (D) = 20 mm
- Heat transfer coefficient for air (h) = 20 W/m²K
- Thermal conductivity of insulation (k) = 0.5 W/mK

Objective:
To determine the thickness of insulation that will result in maximum heat dissipation.

Assumptions:
- Steady-state conditions
- One-dimensional heat transfer
- Uniform heat transfer coefficient and thermal conductivity
- Negligible heat generation within the wire and insulation

Solution:

1. Heat transfer from the wire to the surroundings:
The heat transfer from the wire to the surroundings can be expressed by the following equation:

q = h * A * ΔT

Where,
q = Heat transfer rate (W)
h = Heat transfer coefficient (W/m²K)
A = Surface area of the wire (m²)
ΔT = Temperature difference between the wire and surroundings (K)

2. Surface area of the wire:
The surface area of a wire can be calculated using the formula:

A = π * D * L

Where,
A = Surface area of the wire (m²)
D = Diameter of the wire (m)
L = Length of the wire (m)

3. Heat conduction through the insulation:
The rate of heat conduction through the insulation can be expressed by the following equation:

q = k * A * ΔT / t

Where,
q = Heat transfer rate (W)
k = Thermal conductivity of insulation (W/mK)
A = Surface area of the wire (m²)
ΔT = Temperature difference across the insulation (K)
t = Thickness of insulation (m)

4. Maximizing heat dissipation:
To maximize heat dissipation, we need to minimize the thermal resistance of the system. The thermal resistance of the wire can be expressed as:

R_wire = (1 / h) * (1 / A)

The thermal resistance of the insulation can be expressed as:

R_insulation = (t / k) * (1 / A)

The total thermal resistance of the system can be expressed as:

R_total = R_wire + R_insulation

To minimize the total thermal resistance, we need to minimize the insulation resistance (R_insulation). This can be achieved by maximizing the thickness of insulation (t).

Conclusion:
From the above analysis, it can be concluded that the maximum heat dissipation occurs when the thickness of the insulation is 25 mm (option B). Increasing the thickness of the insulation reduces the thermal resistance of the system, allowing for more efficient heat transfer from the wire to the surroundings.