Animated Solution for Physics - Current Electricity: A $2\text{ W}$ carbon resistor is color coded with green, black, red and brown respectively. The maximum current which can be passed through this resistor is

Imagine you are holding a tiny electronic component—a carbon resistor. It doesn't have its resistance written in plain numbers; instead, it communicates through a universal language of colors. In our problem, the resistor sports four distinct bands: Green, Black, Red, and Brown.

To unlock its secret, we rely on the standard color code chart. The first two bands give us the significant digits. Green stands for $5$, and Black stands for $0$. Together, they form the number $50$.

The third band is the multiplier. Red corresponds to $10^2$ (or $100$). Therefore, the resistance $R$ is calculated as:

The fourth band, Brown, indicates a tolerance of $\pm 1\%$. While crucial for precision circuit design, it doesn't affect our calculation for the maximum current.

Every resistor has a breaking point—a maximum amount of heat it can dissipate before it literally burns up. This is its power rating. Our resistor is rated at $P=2\text{ W}$.

We know from Joule's law of heating that the power dissipated by a resistor when a current $I$ flows through it is given by the master equation:

To find the maximum safe current, we simply rearrange our power equation to solve for $I$:

Now, let's substitute the values we've uncovered:

Simplifying the fraction inside the square root gives us:

Taking the square root is straightforward:

In electronics, we often deal with small currents, so it's customary to express this in milliamperes (mA). By multiplying by $1000$, we get:

And there we have it! The maximum current this resistor can safely handle is $20\text{ mA}$.