Problem 14 - Applied Physics

Question

A step-up transformer on a \(115-V\) line provides a voltage of \(23 \overline{0} 0 \mathrm{~V}\). If the primary coil has \(65.0\) turns, how many turns does the secondary have?

Step-by-Step Solution

Answer

The secondary coil has 1300 turns.

Step 1: Understanding the Transformer Equation

In a transformer, the primary and secondary voltages (V_p,V_s) and the number of turns in the primary and secondary coils (N_p,N_s) are related by the equation:\[ \frac{V_s}{V_p} = \frac{N_s}{N_p} \]This means that the ratio of the secondary voltage to the primary voltage is equal to the ratio of the number of turns in the secondary to the primary.

Step 2: Plugging In Known Values

We know from the problem statement that:- Primary voltage, \( V_p \), is \( 115 \, \text{V} \).- Secondary voltage, \( V_s \), is \( 2300 \, \text{V} \).- Number of turns in primary, \( N_p \), is \( 65.0 \).Substitute these values into the transformer equation:\[ \frac{2300}{115} = \frac{N_s}{65} \]

Step 3: Solving for the Number of Turns in the Secondary

Simplify and solve the equation for \(N_s\):1. Divide \(2300\) by \(115\):\[ \frac{2300}{115} = 20 \]2. Using this in the equation:\[ 20 = \frac{N_s}{65} \]3. Multiply both sides by \(65\) to solve for \(N_s\):\[ N_s = 20 \times 65 = 1300 \]

Step 4: Conclusion

The number of turns in the secondary coil is therefore \( 1300 \).

Key Concepts

Understanding Step-Up Transformers

A step-up transformer is a kind of transformer that increases voltage from the primary side to the secondary side.
In this context, it takes lower voltage input and boosts it to a higher voltage output, which is exactly the scenario given in our original exercise.
These transformers are extensively used in applications where electricity needs to be transmitted over long distances because higher voltage results in lower current, thereby reducing energy loss.
Here's a look at some important points:

• Primary Coil: The side inputting the lower voltage.
• Secondary Coil: The side outputting the higher voltage.
• Step-up transformers have more turns in the secondary coil than the primary coil.

How to Calculate the Voltage Ratio

The voltage ratio of a transformer is crucial because it determines how effectively it can step up or step down voltages.
In order to calculate this ratio, consider the voltages on both sides of the transformer.
This ratio is expressed simply as the voltage of the secondary coil (\( V_s \)) divided by the voltage of the primary coil (\( V_p \)). This fundamental formula looks like this:
\[\frac{V_s}{V_p}\]

• For step-up transformers, this ratio will always be greater than one.
• This means that the voltage in the secondary coil is higher than in the primary coil.
• In our example, the calculation was \( \frac{2300}{115} \), resulting in a ratio of \( 20 \).

Understanding this ratio helps in predicting how much the transformer boosts the voltage.

Determining Transformer Turns

The number of turns in the primary and secondary coils of a transformer gives us insight into how a transformer adjusts voltages.
The relationship between the voltages and the number of turns is directly related.
This means the voltage ratio is equal to the turns ratio, which is calculated using the formula:
\[\frac{N_s}{N_p} = \frac{V_s}{V_p}\]
Where \( N_s \) and \( N_p \) represent the number of turns in the secondary and primary coils respectively.

• For step-up transformers, \( N_s > N_p \).
• In the example given, \( N_s = 1300 \) when \( N_p = 65 \), signifying an increase in turns to raise the voltage.
• Multiply the primary turns by the voltage ratio to find the number of turns in the secondary coil.

This is key in ensuring the transformer performs its role efficiently.

Solving Physics Problems Involving Transformers

Physics problems involving transformers can often be tackled methodically by following a series of logical steps.
Understanding the relationship between voltage and turns is central to solving these problems. Here's a general approach:

• Read the problem carefully and identify known values (e.g., voltages, turns).
• Apply the transformer equation \(\frac{V_s}{V_p} = \frac{N_s}{N_p}\).
• Substitute known values into the equation to find unknown quantities.
• Solve for the unknown, whether it's voltage or number of turns.

In our example, applying these steps allowed us to find the number of turns in the secondary coil.
Such practice not only solidifies understanding but also equips students to handle more complex transformer-related physics scenarios.