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Step-by-Step Solution
Step 1: Understand the Problem
The given ammeter has an internal (or galvanometer) resistance of $13\,\Omega$ and is originally capable of measuring currents up to $100\,\text{A}$. When an external shunt resistance $S$ is connected in parallel with the ammeter, the maximum measurable current increases to $750\,\text{A}$. We need to find the value of this shunt resistor $S$.
Step 2: Express the Key Relationship
An ammeter is essentially a galvanometer in parallel with a shunt resistor. The total current $I$ splits between the galvanometer (which carries a current $I_g$) and the shunt (which carries the rest of the current, $I_S$). For full-scale deflection of the galvanometer:
$I_g = 100\,\text{A} \quad\text{(original maximum current through the meter)}$
When the shunt is added, the total current passing through the combination is $I = 750\,\text{A}$.
The standard formula for the ratio of the total current to the current through the galvanometer is:
$ \displaystyle \frac{I}{I_g} = 1 + \frac{G}{S}$
Here, $G$ is the galvanometer (ammeter's internal) resistance, and $S$ is the shunt resistance we want to determine.
Step 3: Substitute the Given Values
From the problem:
$I = 750\,\text{A}, \quad I_g = 100\,\text{A}, \quad G = 13\,\Omega.
$
So,
$ \displaystyle \frac{750}{100} = 1 + \frac{13}{S}.
$
$ \displaystyle 7.5 = 1 + \frac{13}{S}.
$
Step 4: Solve for the Shunt Resistance $S$
Rearrange to isolate $S$:
$ \displaystyle 7.5 - 1 = \frac{13}{S}.
$
$ \displaystyle 6.5 = \frac{13}{S}.
$
$ \displaystyle S = \frac{13}{6.5} = 2\,\Omega.
$
Step 5: State the Final Answer
The required shunt resistance is therefore $2\,\Omega$.
