Animated Solution for Physics - Current Electricity: A $200\Omega$ resistor has a certain colour code. If one replaces the red colour by green in the code, the new resistance will be

Imagine opening up an old vintage radio or inspecting a modern, high-tech circuit board. Among the dense maze of copper traces, capacitors, and microchips, you will almost certainly spot tiny, cylindrical components adorned with brightly colored stripes. These are resistors, the unsung heroes of electronics that act as the traffic controllers for the flow of electric current.

But why do we use colors instead of just printing the resistance value directly on them? In the early days of electronics, components were incredibly small, and printing technology was not advanced enough to reliably print tiny numbers that could be read from any angle. The solution was brilliantly simple and highly visual: a universal color code. By painting bands of color around the entire circumference of the resistor, engineers ensured that the value could be read easily, no matter how the component was soldered into the circuit.

To master the resistor color code, generations of physics and engineering students have relied on a classic, time-tested mnemonic: BB ROY of Great Britain has a Very Good Wife.

This catchy phrase perfectly maps to the sequence of colors and their corresponding numerical values: - Black (0) - Brown (1) - Red (2) - Orange (3) - Yellow (4) - Green (5) - Blue (6) - Violet (7) - Grey (8) - White (9)

For a standard four-band resistor, the first two bands represent the significant digits of the resistance value. The third band acts as the multiplier, indicating the power of 10 by which the significant digits must be multiplied. The fourth band, usually gold or silver, represents the tolerance or precision of the resistor.

Let us dive into the specific problem at hand. We are given a resistor with an initial resistance of $200\Omega$. Our first task is to decode this numerical value into its corresponding physical color bands.

To do this, we must express the number $200$ in the standard scientific notation format used by the color code system, which is $AB\times 10^C$.

We can rewrite $200$ as $20\times 10^1$.

Now, we can map each part of this mathematical expression to a specific color: - The first significant digit is 2. According to our mnemonic, 2 corresponds to the color Red. - The second significant digit is 0. This corresponds to the color Black. - The multiplier is $10^1$, meaning the power is 1. This corresponds to the color Brown.

Therefore, the original color code of our $200\Omega$ resistor is Red, Black, Brown.

The mathematical backbone of this entire system is the master equation:

Here, $A$ and $B$ are not multiplied together; they are placed side-by-side to form a two-digit number. This equation is the bridge between the physical colors painted on the resistor and the abstract mathematical value of its resistance.

Whenever you face a problem involving resistor color codes, your first instinct should always be to write down this master equation. It provides a clear, structured path from the visual information to the final numerical answer.

Red to Green

The problem introduces a fascinating twist: "If one replaces the red colour by green in the code, what will be the new resistance?"

This is where we must be incredibly careful and analytical. We need to identify exactly which band is being changed and how that affects our master equation.

The original code was Red, Black, Brown. The Red band is the very first band, which represents the first significant digit, $A$.

We are replacing Red with Green. Looking back at our mnemonic, Green corresponds to the number 5.

So, our new first significant digit is 5. The second band (Black, 0) and the third band (Brown, $10^1$) remain completely untouched.

It is worth pausing here to consider a "what if" scenario. What if the problem had asked us to replace the third band (Brown) with Green?

If the multiplier band changed to Green (5), the new multiplier would be $10^5$. The resistance would skyrocket from $200\Omega$ to $20\times 10^5\Omega$, or $2,000,000\Omega$ (2 Megaohms)! This thought experiment highlights how sensitive the resistance value is to the position of the color bands. Changing a significant digit alters the base value linearly, but changing the multiplier alters the value exponentially.

Returning to our actual problem, we simply plug our new values back into the master equation.

Our new significant digits are 5 and 0, forming the number 50. Our multiplier remains $10^1$.

Substituting these into the formula, we get:

Multiplying this out, we arrive at our final answer:

By simply changing the first band from Red to Green, we have increased the resistance from $200\Omega$ to $500\Omega$. This problem beautifully illustrates the elegance and precision of the resistor color code system. A single band change can drastically alter the behavior of an entire electronic circuit. Always read your bands carefully, and trust the math!