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In a transformer, the resistance of the coils plays a crucial role in determining its efficiency and performance. Resistance in electrical circuits is a measure of how much a conductor opposes the flow of electric current. For transformers, coils are essentially long lengths of wire wound into loops. Transformers typically have two coils: the primary and the secondary. The resistance of these coils depends on several factors, including:

• Material of the wire: Copper is commonly used due to its low resistivity.
• Length of the wire: Longer wires have higher resistance.
• Cross-sectional area: Thinner wires have higher resistance for the same length and material.

In the given exercise, the resistance values — 56 Ω for the primary coil and 14 Ω for the secondary coil — indicate that the primary coil offers more resistance than the secondary, largely due to its longer length.

Copper is the most common material used for winding coils in transformers because of its high electrical conductivity and optimum balance of cost and availability. Copper wire resistance increases with the length of the wire and decreases with the increase in its cross-sectional area. When two coils are made from the same type of copper wire with the same diameter, variations in their resistances arise primarily from differences in their lengths. Copper wire resistance is affected by:

• The resistivity of copper, which remains constant under normal working temperatures.
• Uniformity in cross-sectional area, since it remains the same for both coils in this problem.
• Environmental factors like temperature, which can affect the resistance slightly, but under typical conditions, copper's resistivity is stable.

Thus, careful calculations that account for the length differences are necessary to understand each coil's resistance properly, as seen in this exercise.

The concepts of resistivity and resistance are fundamental in electrical engineering and play a significant part in transformer functionality. Resistivity is a material-specific property that quantifies how strongly a given material opposes the flow of electric current. It is denoted by the symbol \( \rho \) and is measured in ohm-meters (Ω·m).The formula to calculate resistance using resistivity is: \[R = \rho \frac{L}{A}\]where:

• \( R \) is the resistance,
• \( \rho \) is the resistivity,
• \( L \) is the length of the wire,
• \( A \) is the cross-sectional area.

Since transformers utilize coils of copper wire, the resistance of these coils is directly influenced by the copper's resistivity, ensuring that longer wires have more resistance, as shown in the primary coil of our example.

The turns ratio in a transformer is vital because it determines the voltage and current transformation between the primary and secondary circuits. It is the ratio of the number of turns in the secondary coil to the number of turns in the primary coil.This can be calculated easily if you know the resistor ratio of the coils along with the type and cross-sectional area of wire used. From the exercise, where both coils are identical in material and diameter, understanding that the primary has four times the resistance of the secondary directly gives us the ratio of lengths of wire used for each coil.Given that length corresponds to turns, we calculate the turns ratio as:\[\frac{N_s}{N_p} = \frac{L_s}{L_p} = \frac{1}{4}\]This turns ratio shows that for every single loop of the copper wire in the secondary coil, there are four in the primary coil. This fundamentally affects the voltage and current levels transformed across the coils, abiding by the principle of energy conservation in electrical transformations.