Animated Solution for Physics - Current Electricity: If $400\Omega$ of resistance is made by adding four $100\Omega$ resistance of tolerance 5%, then the tolerance of the combination is

Imagine you are an electrical engineer tasked with building a highly precise circuit. You need a $400\Omega$ resistor, but all you have in your inventory are $100\Omega$ resistors, each with a tolerance of $5\%$. Naturally, you decide to connect four of these $100\Omega$ resistors in series to achieve your target of $400\Omega$. But a critical question arises: what is the tolerance of this new combination? Will the errors compound and ruin your circuit, or will they behave differently? Let's dive into the fascinating world of error propagation to find out.

When we connect resistors in series, their equivalent resistance is simply the algebraic sum of their individual resistances. Mathematically, this is expressed as:

In our case, $R_1=R_2=R_3=R_4=100\Omega$. Therefore, the equivalent resistance is indeed $400\Omega$. However, the physical world is never perfect. Each resistor has a tolerance of $5\%$, which represents the maximum percentage error in its manufactured value.

According to the fundamental rules of error analysis, when physical quantities are added or subtracted, their absolute errors are added to find the total absolute error. It is a common and fatal mistake to directly add percentage errors.

Let's calculate the absolute error for a single $100\Omega$ resistor. The absolute error $\Delta R$ is $5\%$ of $100\Omega$:

This means each resistor's actual value lies somewhere between $95\Omega$ and $105\Omega$. Since we are adding four such resistors, the total absolute error $\Delta R_{eq}$ will be the sum of their individual absolute errors:

Now we have the total equivalent resistance ($400\Omega$) and the total absolute error ($20\Omega$). The question asks for the tolerance of the combination, which is the percentage error of the equivalent resistance. We can find this by dividing the total absolute error by the total resistance and multiplying by $100$:

Notice something magical? We started with individual resistors having a $5\%$ tolerance, and after combining four of them, the overall tolerance is still exactly $5\%$.

This reveals a beautiful and highly useful shortcut for competitive exams like JEE: Whenever you combine $n$ identical components (whether in series or parallel) that have the same percentage error, the overall percentage error of the combination remains completely unchanged.

The absolute error scales up by a factor of $n$, but because the base value also scales up by the exact same factor of $n$, the ratio (and thus the percentage) remains constant. Understanding this intuitive principle not only saves precious time during an exam but also deepens your appreciation for how uncertainties behave in physical systems.