Ammeter Conversion: Calculating Shunt Resistance

This problem requires us to determine the value of a shunt resistance needed to extend the range of an ammeter. An ammeter, at its core, is often a galvanometer. To measure currents larger than its original full-scale deflection current, a small resistance, called a shunt resistance (\(R_s\)), is connected in parallel with the ammeter.

When a shunt resistance is connected in parallel, the total current (\(I\)) that enters the combination splits. A small portion of the current, corresponding to the original full-scale deflection current (\(I_g\)), passes through the ammeter (galvanometer resistance, \(R_g\)), while the major portion of the current passes through the shunt resistance.

Since the ammeter and the shunt resistance are in parallel, the voltage drop across them is the same. Let \(V\) be the voltage drop across the parallel combination. Voltage drop across the ammeter = \(I_g \times R_g\) Voltage drop across the shunt = \((I - I_g) \times R_s\)

Equating the voltage drops:

$$I_g R_g = (I - I_g) R_s$$

We need to calculate the shunt resistance \(R_s\). Rearranging the formula to solve for \(R_s\), we get:

$$R_s = \frac{I_g R_g}{I - I_g}$$

Applying Given Values to Calculate Shunt Resistance

We are given the following parameters:

• Full-scale deflection current of the ammeter, \(I_g = 1 \, \text{mA}\)
• Resistance of the ammeter, \(R_g = 100 \, \Omega\)
• Desired total current range for the converted ammeter, \(I = 1 \, \text{A}\)

First, we need to ensure all current values are in the same units (Amperes). $$I_g = 1 \, \text{mA} = 1 \times 10^{-3} \, \text{A} = 0.001 \, \text{A}$$

Now, we can substitute these values into the formula for \(R_s\):

$$R_s = \frac{(0.001 \, \text{A}) \times (100 \, \Omega)}{1 \, \text{A} - 0.001 \, \text{A}}$$

Calculate the numerator:

$$0.001 \times 100 = 0.1$$

Calculate the denominator:

$$1 - 0.001 = 0.999$$

Now, calculate \(R_s\):

$$R_s = \frac{0.1}{0.999} \, \Omega$$

Performing the division:

$$R_s \approx 0.1001001... \, \Omega$$

Rounding the result to four decimal places, we get \(R_s \approx 0.1001 \, \Omega\).

Comparing Calculation with Options

Let's compare our calculated value of approximately \(0.1001 \, \Omega\) with the given options:

• Option 1: \(10000 \, \Omega\)
• Option 2: \(0.1001 \, \Omega\)
• Option 3: \(0.01 \, \Omega\)
• Option 4: \(1000 \, \Omega\)

Our calculated value closely matches Option 2.

Revision Table: Ammeter and Voltmeter Conversion Formulas

Additional Information: Role of Shunt Resistance

The shunt resistance in an ammeter serves a crucial purpose. By providing an alternative, low-resistance path for most of the current, it protects the sensitive galvanometer coil from damage due to excessive current. Only a small, proportional fraction of the total current passes through the galvanometer, which is within its safe operating range. The parallel combination of the galvanometer and the shunt resistance behaves as an ammeter with an extended range. The total resistance of this ammeter is very low, which is a desirable characteristic for an ammeter as it should ideally have zero resistance to avoid affecting the circuit it is measuring current in.