Problem 62 - Fundamentals of Physics

Question

A generator supplies \(100 \mathrm{~V}\) to a transformer's primary coil, which has 50 turns. If the secondary coil has 500 turns, what is the secondary voltage?

Step-by-Step Solution

Answer

The secondary voltage is 1000 V.

Step 1: Understand the Transformer Formula

A transformer operates on the principle of electromagnetic induction and the relationship between the primary and secondary voltages, and the number of turns is given by the formula \( \frac{V_s}{V_p} = \frac{N_s}{N_p} \), where \( V_s \) and \( V_p \) are the secondary and primary voltages respectively, and \( N_s \) and \( N_p \) are the number of turns on the secondary and primary coils.

Step 2: Identify Known Values

From the problem statement, identify the known values: \( V_p = 100 \mathrm{~V} \), \( N_p = 50 \), and \( N_s = 500 \). The task is to find the secondary voltage \( V_s \).

Step 3: Write the Transformer Equation

Using the transformer formula \( \frac{V_s}{V_p} = \frac{N_s}{N_p} \), substitute the known values: \( \frac{V_s}{100} = \frac{500}{50} \).

Step 4: Simplify the Right Side

The right side of the equation simplifies to \( \frac{500}{50} = 10 \). The equation becomes \( \frac{V_s}{100} = 10 \).

Step 5: Solve for Secondary Voltage

Multiply both sides of the equation \( \frac{V_s}{100} = 10 \) by 100 to isolate \( V_s \). This gives \( V_s = 100 \times 10 = 1000 \mathrm{~V} \).

Step 6: Conclusion

The secondary voltage is \( 1000 \mathrm{~V} \).

Key Concepts

Electromagnetic Induction

Electromagnetic induction is a fundamental principle allowing a transformer to operate. When an electric current passes through a coil, it creates a magnetic field around that coil, and if there's another coil nearby placed within this magnetic field, an electrical current is induced in the second coil. This process enables the transfer of energy from one circuit to another without direct contact or connectors. In transformers, this is used to step up or step down voltages between the primary and secondary coils. This dynamic interaction makes transformers vital components in electrical power systems for efficiently modifying voltage levels.

Transformer Formula

The crucial formula used in transformers is \( \frac{V_s}{V_p} = \frac{N_s}{N_p} \), where:

• \( V_s \) is the secondary voltage
• \( V_p \) is the primary voltage
• \( N_s \) is the number of turns in the secondary coil
• \( N_p \) is the number of turns in the primary coil

This formula is derived from the principle of electromagnetic induction and expresses the direct correlation between the number of coil turns and the voltage across them. By applying this equation, you can calculate unknown values, like the secondary voltage when the other three parameters are known, making it foundational for solving transformer-related problems.

Voltage Transformation

Transformers efficiently change voltage levels from one magnitude to another using the principle of voltage transformation. When voltage is increased by the transformer, it is termed as 'stepping up', and when the voltage is decreased, it is referred to as 'stepping down'. The actual transformation is governed by the coil turns ratio, which determines how the primary voltage is translated to the secondary voltage. This transformation process is key in adapting power levels to appropriate scales required for different applications. For instance, in the original exercise, a primary voltage of 100 V is transformed into a secondary voltage of 1000 V. Such transformations are crucial for transmitting electrical power across long distances efficiently, reducing power losses before it's distributed for local usage.

Coil Turns Ratio

The coil turns ratio is the ratio of the number of turns in the secondary coil to the number of turns in the primary coil. It directly affects how the input voltage is transformed into output voltage in a transformer. The formula \( \frac{N_s}{N_p} \) gives this turns ratio and shows how directly proportional it is to the voltage transformation.In our exercise, the turns ratio is \( \frac{500}{50} = 10 \). This means the secondary voltage will be ten times higher than the primary voltage, given all other conditions remain constant. Understanding this simple yet powerful aspect of transformers helps in designing and utilizing them effectively in various electrical systems.