The primary windings of a transformer which has an 85% efficiency are connected to 110-V ac. The secondary windings are connected across a 2.4-Ω, 75-W lightbulb.
(a) Calculate the current through the primary windings of the transformer.
(b) Calculate the ratio of the number of primary windings of the transformer to the number of secondary windings of the transformer.

Verified step by step guidance

1. Step 1: Start by understanding the efficiency of the transformer. The efficiency (η) is given as 85%, which means that 85% of the power input to the primary windings is transferred to the secondary windings. Use the formula for efficiency: η = (P_secondary / P_primary) × 100, where P_secondary is the power delivered to the load and P_primary is the power input to the primary windings.
2. Step 2: Calculate the power delivered to the load (P_secondary). The lightbulb connected to the secondary windings has a resistance of 2.4 Ω and a power rating of 75 W. Since the lightbulb operates at its rated power, the voltage across the secondary windings (V_secondary) can be calculated using the formula P = V² / R. Rearrange to find V_secondary: V_secondary = √(P × R).
3. Step 3: Use the efficiency formula to calculate the power input to the primary windings (P_primary). Rearrange the efficiency formula to find P_primary: P_primary = P_secondary / η. Substitute the known values of P_secondary and η (converted to decimal form, 0.85).
4. Step 4: Calculate the current through the primary windings (I_primary). The voltage across the primary windings (V_primary) is given as 110 V. Use the formula P = IV to find I_primary: I_primary = P_primary / V_primary. Substitute the values of P_primary and V_primary to find I_primary.
5. Step 5: Calculate the turns ratio of the transformer. The turns ratio (N_primary / N_secondary) is related to the voltage ratio by the formula V_primary / V_secondary = N_primary / N_secondary. Rearrange to find the turns ratio: N_primary / N_secondary = V_primary / V_secondary. Substitute the values of V_primary and V_secondary to calculate the ratio.

Key Concepts

Transformer Basics

A transformer is an electrical device that transfers electrical energy between two or more circuits through electromagnetic induction. It consists of primary and secondary windings, where the primary winding receives input voltage and the secondary winding delivers output voltage. The voltage transformation ratio is determined by the ratio of the number of turns in the primary and secondary coils.

Efficiency of a Transformer

The efficiency of a transformer is the ratio of the output power to the input power, expressed as a percentage. In this case, an 85% efficiency means that 85% of the input electrical energy is converted into useful output energy, while the remaining 15% is lost, typically as heat. This efficiency is crucial for calculating the current and power in both primary and secondary circuits.

Ohm's Law and Power Calculation

Ohm's Law states that the current through a conductor between two points is directly proportional to the voltage across the two points and inversely proportional to the resistance. This relationship is essential for calculating the current through the lightbulb and the primary windings. Additionally, power can be calculated using the formula P = IV, where P is power, I is current, and V is voltage.