Problem 107 - Physics

Question

A galvanometer with a coil resistance of \(12.0 \Omega\) and a full-scale current of \(0.150 \mathrm{~mA}\) is used with a shunt resistor to make an ammeter. The ammeter registers a maximum current of \(4.00 \mathrm{~mA}\). Find the equivalent resistance of the ammeter.

Step-by-Step Solution

Answer

The equivalent resistance of the ammeter is approximately \(0.438 \Omega\).

Step 1: Determine Shunt Resistance

The shunt resistor is used to divert most of the current away from the galvanometer. First, calculate the shunt resistance using the formula: \( R_s = \frac{R_g \cdot I_g}{I - I_g} \), where \( R_g = 12.0 \Omega \) is the galvanometer resistance, \( I_g = 0.150 \text{ mA} \) is the full-scale current of the galvanometer, and \( I = 4.00 \text{ mA} \) is the total ammeter current. Substituting these values, we have: \( R_s = \frac{12.0 \cdot 0.150}{4.00 - 0.150} \approx 0.455 \Omega \).

Step 2: Calculate Equivalent Resistance of the Ammeter

The equivalent resistance of the ammeter is the combination of the galvanometer resistance \( R_g \) in parallel with the shunt resistor \( R_s \). Use the parallel resistance formula: \( R_a = \frac{R_g \cdot R_s}{R_g + R_s} \). Substituting \( R_g = 12.0 \Omega \) and \( R_s \approx 0.455 \Omega \), we have: \( R_a = \frac{12.0 \cdot 0.455}{12.0 + 0.455} \approx 0.438 \Omega \).

Key Concepts

Shunt Resistor

A shunt resistor is a very handy device used in electrical circuits to allow a portion of current to bypass a main component, like a galvanometer, when measuring larger currents. By connecting a shunt resistor in parallel with the galvanometer, the sensitive instrument is protected from large currents that could damage it. This setup enables us to extend the measurement range of the galvanometer to measure higher currents and effectively turn it into an ammeter.

• Formula: The shunt resistance \( R_s \) is calculated using the formula: \( R_s = \frac{R_g \cdot I_g}{I - I_g} \).
• This formula helps determine how much resistance is needed to safely divert the excess current.
• In practical terms, the shunt resistor has a small resistance value compared to the galvanometer's resistance, ensuring most of the current goes through it.

When implemented correctly, this arrangement allows for accurate current measurement while keeping the delicate galvanometer safe from being overloaded.

Equivalent Resistance

The equivalent resistance in the context of a parallel circuit is the total resistance faced by current when multiple resistors share paths. Calculating this is crucial when using a shunt resistor with a galvanometer to form an ammeter. In this setup, the galvanometer's resistance \( R_g \) and the shunt resistor \( R_s \) form a combined effective resistance known as the equivalent resistance \( R_a \).

• To find this, use the formula \( R_a = \frac{R_g \cdot R_s}{R_g + R_s} \).
• This reflects the resistance seen by the entire current, combining the pathways provided by the galvanometer and shunt resistor.
• In the example exercise, the computed equivalent resistance is approximately \(0.438 \Omega\).

Understanding equivalent resistance is important for ensuring the measuring setup doesn't significantly affect the circuit's overall resistance. This prevents the altering of current levels which can lead to inaccurate readings.

Ammeter

An ammeter is an instrument used to measure electric current in a circuit. By converting a galvanometer into an ammeter using a shunt resistor, one can measure a much larger range of currents than the galvanometer alone could handle. This transformation is essential to allowing accurate current measurements across various electrical setups.

• The ammeter has a special property: it must have a very low equivalent resistance, so it doesn’t change the current it is meant to measure.
• This is achieved by correctly choosing a shunt resistor to bypass most of the current.
• In our solved exercise, the chosen shunt ensures the ammeter registers up to \(4.00 \,\mathrm{mA}\), beyond the galvanometer's initial \(0.150 \,\mathrm{mA}\).

When constructing or using an ammeter, it is vital to ensure the instrument remains efficient, reliable, and offers simple integration into various circuits. With these elements in mind, an ammeter effectively assists engineers and technicians in monitoring current flow safely in numerous applications.