Problem 15 - Applied Physics

Question

A step-down transformer on a \(115-V\) line provides a voltage of \(11.5 \mathrm{~V}\). If the secondary coil has \(30.0\) turns, how many turns does the primary have?

Step-by-Step Solution

Answer

The primary coil has 300 turns.

Step 1: Understanding the Problem

We have a step-down transformer with a primary voltage of 115 V and a secondary voltage of 11.5 V. We need to determine the number of turns in the primary coil given there are 30.0 turns in the secondary coil.

Step 2: Applying the Transformer Equation

The transformer equation relates the number of turns in the coils to the voltages: \( \frac{V_p}{V_s} = \frac{N_p}{N_s} \), where \( V_p \) and \( V_s \) are the primary and secondary voltages, and \( N_p \) and \( N_s \) are the number of turns in the primary and secondary coils, respectively. We know \( V_p = 115 \) V, \( V_s = 11.5 \) V, and \( N_s = 30 \). We need to find \( N_p \).

Step 3: Solving for the Primary Turns

Rearrange the transformer equation to solve for \( N_p \):\[ N_p = \frac{V_p}{V_s} \times N_s \]Substitute the known values:\[ N_p = \frac{115}{11.5} \times 30 \]

Step 4: Calculating the Result

Perform the calculations:\[ \frac{115}{11.5} = 10 \]Now multiply by the number of turns in the secondary:\[ N_p = 10 \times 30 = 300 \]

Step 5: Conclusion

The number of turns in the primary coil is 300.

Key Concepts

Step-Down Transformer

A step-down transformer is a device that reduces high voltage to a lower voltage. It is commonly used to decrease the voltage from power lines to a level suitable for household or industrial use.

• Primary Coil: This coil receives the input voltage, which is higher in step-down transformers.
• Secondary Coil: This coil delivers the output voltage, which is lower than the primary.
• Core: Made of iron or similar materials, the core helps guide the magnetic field between the coils.

The mechanism of a step-down transformer relies on electromagnetic induction. When an alternating current flows through the primary coil, it creates a changing magnetic field. This induced magnetic field generates a current in the secondary coil, having a lower voltage due to the coil's fewer turns. Thus, the primary function of a step-down transformer is to adjust the voltage while conserving the power transmitted in the process.

Transformer Turns Calculation

In transformers, the number of turns in each coil affects the voltage transformation. This relationship is depicted by the transformer equation: \[ \frac{V_p}{V_s} = \frac{N_p}{N_s} \]

• \(V_p\): Voltage across the primary coil.
• \(V_s\): Voltage across the secondary coil.
• \(N_p\): Number of turns in the primary coil.
• \(N_s\): Number of turns in the secondary coil.

Given this relationship, you can calculate the unknown number of turns in one coil if the other parameters are known. In a step-down transformer, for instance, if we know the secondary coil has 30 turns and the secondary voltage is significantly reduced compared to the primary, the turns on the primary coil must be higher to maintain the equation's balance. This ensures that an increase or decrease in voltage is directly proportional to the respective increase or decrease in the number of coil turns.

Primary and Secondary Voltage Relationship

In a transformer, the relationship between primary and secondary voltage is crucial for its operation. The fundamental aspect of this relationship is the inverse proportionality between voltage and the number of coil turns, as described by \[ \frac{V_p}{V_s} = \frac{N_p}{N_s} \]

• If the primary coil has more turns than the secondary, the transformer steps the voltage down, reducing \(V_s\).
• If the secondary coil has more turns, the transformer steps the voltage up, increasing \(V_s\).

This relationship ensures energy conservation, as transforms alter voltage at the cost of current. For step-down transformers, like the one in the provided exercise, \(V_p\) (initial high voltage) will be greater than \(V_s\) (final reduced voltage). Calculating the respective turns and confirming the transformer equation assures that energy input and output maintain equilibrium, intuitive for both efficiency and safety.