Problem 65 - Master Resource Book in JEE Main Physics

Question

A transformer is having 2100 turns in primary and 4200 turns in secondary. An AC source of \(120 \mathrm{~V}, 10 \mathrm{~A}\) is connected to its primary. The secondary voltage and current are (a) \(240 \mathrm{~V}, 5 \mathrm{~A}\) (b) \(120 \mathrm{~V}, 10 \mathrm{~A}\) (c) \(240 \mathrm{~V}, 10 \mathrm{~A}\) (d) \(120 \mathrm{~V}, 20 \mathrm{~A}\)

Step-by-Step Solution

Answer

The secondary voltage and current are 240 V and 5 A; option (a) is correct.

Step 1: Understanding the Transformer Ratio

Transformers work based on the principle of electromagnetic induction and the voltage ratio across the transformer is directly proportional to the turn ratio. The formula for the transformer turn ratio 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, and \(N_p\) is the number of turns in the primary. Here, \(N_p = 2100\) turns, \(N_s = 4200\) turns, and \(V_p = 120\, \text{V}.\)

Step 2: Calculating the Secondary Voltage

Using the transformer turn ratio formula, we can calculate the secondary voltage \(V_s\) as follows: \[ V_s = \frac{N_s}{N_p} \times V_p = \frac{4200}{2100} \times 120 = 2 \times 120 = 240 \, \text{V}.\] So, the secondary voltage is 240 V.

Step 3: Applying the Power Conservation Principle

In an ideal transformer, the power in the primary coil is equal to the power in the secondary coil. Therefore, \(V_p \times I_p = V_s \times I_s\), where \(I_p\) is the primary current and \(I_s\) is the secondary current. Given that \(V_p = 120\, \text{V}\) and \(I_p = 10\, \text{A}\), the power is \(120 \times 10 = 1200 \, \text{W}.\) With \(V_s = 240\, \text{V}\), we can solve for \(I_s:\) \[ 240 \times I_s = 1200\] \[ I_s = \frac{1200}{240} = 5 \, \text{A}.\] Hence, the secondary current is 5 A.

Step 4: Selecting the Correct Option

Based on our calculations, the secondary voltage is 240 V and the current is 5 A. Therefore, the correct option that matches these values is \((a)\ \text{240 V, 5 A}.\)

Key Concepts

Electromagnetic Induction

Electromagnetic induction is a fundamental principle that allows transformers to function. When an alternating current (AC) flows through the primary coil, it creates a magnetic field. This changing magnetic field induces a voltage in the secondary coil through electromagnetic induction. The key takeaway here is the importance of the alternating nature of the current.
Transformers rely on this changing current to continuously create a varying magnetic field, crucial for inducing voltage. Without the principle of electromagnetic induction, the energy transfer from the primary to the secondary coil would not be possible.
This concept is critical not only in transformers but also in many other electrical devices that rely on AC, such as generators and inductors.

Turns Ratio

The turns ratio of a transformer is a vital component that determines the relationship between the voltages across the primary and secondary coils. The formula for the turns ratio is:

• \( \frac{V_s}{V_p} = \frac{N_s}{N_p} \)

\(V_s\) is the secondary voltage, \(V_p\) is the primary voltage, \(N_s\) is the number of turns in the secondary coil, and \(N_p\) is the number of turns in the primary coil.
By understanding the turns ratio, you can predict how much the voltage will increase or decrease. For instance, in the provided exercise, with a primary turn of 2100 and secondary of 4200, the turns ratio is 2:1. Hence, the secondary voltage is twice that of the primary.
Be mindful of this ratio; when designing or examining transformers, it helps ensure devices achieve the desired voltage transformation efficiently.

Power Conservation in Transformers

Power conservation in transformers is about maintaining balance. In an ideal transformer, the power input to the primary coil equals the power output from the secondary coil. This is because transformers cannot create energy; they merely pass it along from one set of coils to another.
The principle of power conservation is expressed with the equation:

• \( V_p \times I_p = V_s \times I_s \)

Here, \(V_p\) and \(I_p\) are the primary voltage and current, whereas \(V_s\) and \(I_s\) are the secondary voltage and current.
Despite real-world resistive losses, which cause a slight deviation from ideality, this equation is a foundational concept for understanding how energy transitions efficiently through transformers.
By preserving power, transformers optimize energy transfer, ensuring devices connected downstream receive the correct power levels for operation.