Transformers: Practice Problem & Solution
A transformer is used to light a 100 W and 110 V lamp from a 220 V mains. If the main current is 0.5 amp, the efficiency of the transformer is approximately: (2007)
Short Solution (Best for NEET):
To solve this problem, we apply the core principles of Transformers. Understanding the underlying formula is key to arriving at the correct answer below:
Output power is $P_{out} = 100 W$.
Input power is $P_{in} = V_p I_p = 220 \times 0.5 = 110 W$.
Efficiency $\eta = \frac{P_{out}}{P_{in}} = \frac{100}{110} = 0.909$, which is approximately $90\%$.
Detailed Solution (Best of Understanding Complex Concepts)
Concepts Involved
-
Transformer Principle: A transformer transfers electrical energy from one alternating-current circuit to another through electromagnetic induction. It is used to step up (increase) or step down (decrease) voltage.
-
Electrical Power: The rate at which electrical energy is transferred by an electric circuit. The input power for the primary coil is calculated as $P_{in} = V_p I_p$, where $V_p$ is the primary voltage and $I_p$ is the primary current.
-
Transformer Efficiency ($\eta$): The ratio of useful power output (secondary coil) to the total power input (primary coil). It is typically expressed as a percentage: $\eta = \left( \frac{P_{out}}{P_{in}} \right) \times 100\%$.
Step-by-Step Solution
Step 1: Identify the given information.
-
Output Power ($P_{out}$): The power consumed by the lamp connected to the secondary coil is $100\text{ W}$. (Note: The 110 V rating of the lamp tells us the secondary voltage, but since the output power is already provided, we don't need to use it for this calculation).
-
Input Voltage ($V_{in}$ or $V_p$): The mains voltage is $220\text{ V}$.
-
Input Current ($I_{in}$ or $I_p$): The current drawn from the mains is $0.5\text{ A}$.
Step 2: Calculate the Input Power ($P_{in}$).
The power drawn from the mains by the primary coil of the transformer is:
$$P_{in} = V_{in} \times I_{in}$$
$$P_{in} = 220 \times 0.5$$
$$P_{in} = 110\text{ W}$$
Step 3: Calculate the Efficiency ($\eta$).
Efficiency is the ratio of output power to input power:
$$\eta = \frac{P_{out}}{P_{in}} \times 100\%$$
$$\eta = \frac{100}{110} \times 100\%$$
$$\eta = \frac{10}{11} \times 100\%$$
$$\eta \approx 90.9\%$$
The efficiency of the transformer is approximately 90%.
Leave a Reply