The mathematical problem involves calculating the ratio of currents in an electrical transformer using the transformer equations, specifically the turns ratio and impedance relationships.

Answer:
The ratio of the secondary to primary current is \(\frac{I_{2}}{I_{1}} = \frac{Z_{1}E_{2}^{2}}{Z_{2}E_{1}^{2}} = \frac{4n}{(1 + n)^{2}}\).

Explanation:
This problem applies the principles of transformer theory, including the relationships between voltages, currents, and impedances. It uses the concept that the current ratio is inversely proportional to the turns ratio, adjusted by the impedance and voltage relationships. The derivation involves the use of the transformer equations and the concept of impedance transformation.

Steps:

1. Transformer voltage and current relationships:

The ratio of the number of turns \(n\) relates to the voltage and impedance ratios.

1. Current ratio in terms of impedance and voltage:

This is derived from the transformer equations, considering the voltage transformation ratio.

1. Expressing the ratio of currents:

where \(Z_{i}\) and \(Z_{t}\) are the impedances, and \(E_{t}\) is the voltage.

1. Final expression:

This relates the secondary current to the primary current, scaled by the impedance and voltage ratios.

Summary:
The key concepts involved are the transformer equations, impedance transformation, and the relationships between voltage, current, and impedance ratios in an ideal transformer. The derivation uses algebraic manipulation of these relationships to arrive at the expression for the current ratio.