To solve the problem of how many turns the secondary coil of an ideal transformer should have to produce an output voltage of 30 V when the primary coil has 100 turns and is connected to a 120 V AC source, we can use the transformer equation. ### Step-by-Step Explanation 1. **Understanding the Transformer Equation**: The relationship between the primary voltage (\(V_p\)), secondary voltage (\(V_s\)), the number of turns in the primary coil (\(N_p\)), and the number of turns in the secondary coil (\(N_s\)) is given by the formula: \[ \frac{V_p}{V_s} = \frac{N_p}{N_s} \] This equation tells us that the ratio of the primary voltage to the secondary voltage is equal to the ratio of the number of turns in the primary coil to the number of turns in the secondary coil. 2. **Identifying Known Values**: - Primary voltage (\(V_p\)) = 120 V - Secondary voltage (\(V_s\)) = 30 V - Number of turns in the primary coil (\(N_p\)) = 100 turns 3. **Setting Up the Equation**: We need to find the number of turns in the secondary coil (\(N_s\)). Rearranging the transformer equation gives us: \[ N_s = N_p \cdot \frac{V_s}{V_p} \] 4. **Substituting Known Values**: Now we can substitute the known values into the equation: \[ N_s = 100 \cdot \frac{30}{120} \] 5. **Calculating the Output**: First, simplify the fraction: \[ \frac{30}{120} = \frac{1}{4} \] Now substitute this back into the equation: \[ N_s = 100 \cdot \frac{1}{4} = 25 \] 6. **Final Answer**: Therefore, the number of turns in the secondary coil should be **25 turns**. ### Why the Other Options Are Incorrect - **Option B: 50 turns**: If the secondary had 50 turns, the output voltage would be calculated as: \[ V_s = V_p \cdot \frac{N_s}{N_p} = 120 \cdot \frac{50}{100} = 60 \text{ V} \] This is higher than the desired 30 V. - **Option C: 75 turns**: Similarly, if the secondary had 75 turns: \[ V_s = 120 \cdot \frac{75}{100} = 90 \text{ V} \] Again, this is too high. - **Option D: 100 turns**: If the secondary had 100 turns: \[ V_s = 120 \cdot \frac{100}{100} = 120 \text{ V} \] This would mean the secondary voltage is the same as the primary voltage, which is not what we want. ### Summary of Key Points - The transformer equation relates the voltages and turns of the coils. - For a primary voltage of 120 V and a desired secondary voltage of 30 V, the secondary coil must have 25 turns. - The calculations show that other options yield voltages that do not meet the requirement. ### Revision Summary - Use the transformer equation: \(\frac{V_p}{V_s} = \frac{N_p}{N_s}\). - Rearrange to find \(N_s\): \(N_s = N_p \cdot \frac{V_s}{V_p}\). - Substitute known values carefully to avoid calculation errors. - Verify that the calculated turns produce the desired voltage.