Animated Solution for Physics - Electromagnetic Induction: A power transmission line feeds input power at $2300\text{ V}$ to a step-down transformer with its primary windings having $4000$ turns. The output power is delivered at $230\text{ V}$ by the transformer. If the current in the primary of the transformer is $5\text{ A}$ and its efficiency is $90\%$, the output current would be

Transformers are the unsung heroes of our modern electrical grid. They allow us to transmit power over vast distances at incredibly high voltages to minimize energy loss, and then safely step that voltage down before it enters our homes. But this process isn't perfectly magical—there are always some energy losses involved. Let's dive into the physics of a real-world step-down transformer and see how efficiency dictates the final output.

In an ideal world, a transformer would transfer $100\%$ of the electrical power from its primary coil to its secondary coil. However, real transformers experience losses due to factors like eddy currents, magnetic hysteresis, and the resistance of the copper windings.

To account for this, we use the concept of efficiency ($\eta$), which is simply the ratio of the useful output power to the total input power, expressed as a percentage:

Since electrical power is the product of voltage and current ($P=VI$), we can rewrite this master equation specifically for our transformer:

Here, $V_s$ and $I_s$ are the secondary (output) voltage and current, while $V_p$ and $I_p$ are the primary (input) voltage and current.

Let's extract the vital statistics from our problem. We are given: - Primary Voltage, $V_p=2300\text{ V}$ - Primary Current, $I_p=5\text{ A}$ - Secondary Voltage, $V_s=230\text{ V}$ - Efficiency, $\eta =90\%$

You might notice that the problem also mentions the primary coil has $4000$ turns ($N_p=4000$). In physics problems, examiners often throw in extra information to test your conceptual clarity. Since we already have the voltages and the efficiency, we don't actually need the number of turns to find the output current!

Let's substitute our known values into the efficiency equation:

Now, it's just a matter of careful algebraic manipulation. Let's simplify the fraction on the right side. Notice the elegant relationship between the voltages:

Substituting this back into our equation makes the math much cleaner:

Finally, dividing both sides by $2$ gives us our target variable:

Take a moment to appreciate the result. The input current was a mere $5\text{ A}$, but the output current is a massive $45\text{ A}$. This is the hallmark of a step-down transformer: as it steps the voltage down (from $2300\text{ V}$ to $230\text{ V}$), it must step the current up to conserve power.

If this transformer were $100\%$ efficient, the output current would have been exactly $50\text{ A}$. However, because $10\%$ of the input power is lost as heat, the secondary coil can only deliver $45\text{ A}$ to the load. Always ensure your final mathematical answer aligns with the physical reality of the system!