Explanation:

To maintain the same no-load current, we need to keep the magnetic flux in the core of the transformer constant. The magnetic flux is given by the product of the magnetic field strength (H) and the cross-sectional area of the core (A). The field strength is determined by the number of turns in the primary winding (N) and the input current (I) while the cross-sectional area is determined by the equivalent radius of the core (r).

Step 1: Determine the relationship between the magnetic flux and the number of turns:

The magnetic flux (Φ) is given by the equation Φ = H * A. Rearranging this equation, we have A = Φ / H.

Step 2: Determine the relationship between the field strength and the number of turns:

The field strength (H) is given by the equation H = N * I / l, where l is the length of the magnetic path. Rearranging this equation, we have N = H * l / I.

Step 3: Combine the equations:

Substituting the value of H from the second equation into the first equation, we can express the cross-sectional area (A) in terms of N, I, and r as A = (Φ / (N * I / l)).

Step 4: Determine the relationship between the cross-sectional area and the equivalent radius:

The cross-sectional area of a circular core is given by the equation A = π * r^2. Rearranging this equation, we have r = sqrt(A / π).

Step 5: Combine the equations:

Substituting the value of A from the third equation into the fourth equation, we can express the equivalent radius (r) in terms of N, I, and l as r = sqrt((Φ / (N * I / l)) / π).

Step 6: Analyze the given scenario:

In the given scenario, the equivalent radius is reduced by half. Therefore, the new radius (r') is given by r' = r / 2.

Step 7: Determine the relationship between the new radius and the number of turns:

Substituting the value of r' into the equation from step 5, we have r' = sqrt((Φ / (N' * I / l)) / π), where N' is the new number of turns.

Squaring both sides of the equation, we get r'^2 = (Φ / (N' * I / l)) / π.

Rearranging this equation, we have N' = Φ / (r'^2 * I / l * π).

Step 8: Compare the original and new number of turns:

Comparing the equations for the original and new number of turns, we have N' = Φ / (r'^2 * I / l * π) = Φ / (r^2 / 4 * I / l * π) = 4 * Φ / (r^2 * I / l * π) = 4 * N.

Therefore, the factor by which the number of turns in the primary should change to maintain the same no-load current is 4, which corresponds to option D.