Question

A galvanometer having a resistance of 25.0 $\mathrm{\Omega }$ has a 1.00-$\mathrm{\Omega }$ shunt resistance installed to convert it to an ammeter. It is then used to measure the current in a circuit consisting of a 15.0-$\mathrm{\Omega }$ resistor connected across the terminals of a 25.0-V battery having no appreciable internal resistance. (a) What current does the ammeter measure? (b) What should be the $true$ current in the circuit (that is, the current without the ammeter present)? (c) By what percentage is the ammeter reading in error from the $true$ current?

Short answer

(a) The ammeter measures 22.84 A. (b) The true current is 1.67 A. (c) The reading error is 1273%. A detailed step-by-step solution will follow.

Step-by-step solution

Calculate Total Resistance of the Ammeter

The galvanometer and shunt resistance are in parallel. Use the formula for parallel resistors: $$R_{amm}=\frac{R_g\cdot R_s}{R_g+R_s}$$where $R_g=25.0\mathrm{\Omega }$ and $R_s=1.00\mathrm{\Omega }$. Substitute the values to find:$$R_{amm}=\frac{25.0\cdot 1.00}{25.0+1.00}=\frac{25.0}{26.0}=0.9615\mathrm{\Omega }$$.

Key concepts

Ammeter Measurement

An ammeter is a crucial tool for measuring the current flowing through a circuit. It operates by being connected in series with the circuit, which means it must measure the full current without altering it significantly. However, an ammeter cannot be used directly; it needs to be calibrated or adjusted using other components such as a galvanometer.

A galvanometer is a sensitive instrument that can detect small currents and needs to be converted to function as an ammeter. This conversion is done by adding a component called a shunt resistor. The shunt resistor is connected in parallel to the galvanometer, allowing most of the current to pass through the shunt resistor, thereby protecting the galvanometer from high currents that could damage it.

The resistance of the shunt limits the amount of current measured, ensuring that the galvanometer remains accurate and not overloaded when larger currents flow through the circuit.

Parallel Resistors

In electrical circuits, components like resistors can be connected in series or parallel. When resistors are connected in parallel, they provide multiple paths for the current to flow. This is especially useful in converting sensitive instruments.

The total resistance ($R_{total}$) of resistors in parallel is given by the formula:$$R_{total}=\frac{R_1\cdot R_2}{R_1+R_2}$$This formula is derived from the fact that the voltage across each resistor is the same, yet the total current is equal to the sum of the current through each resistor.

In our case, a galvanometer with resistance$R_g$ of 25.0$\mathrm{\Omega }$ is connected in parallel with a shunt resistor ($R_s$) of 1.00$\mathrm{\Omega }$. By applying the parallel resistor formula, we found the effective resistance ($R_{amm}$) of the combined setup to be approximately 0.9615$\mathrm{\Omega }$, enabling us to measure higher currents without damaging the galvanometer.

Circuit Current Calculation

Calculating the current in a circuit is a key step in electrical analysis, often relying on Ohm's Law:$V=IR$, where$V$ is voltage,$I$ is current, and$R$ is resistance. When an ammeter is correctly used, it measures the actual current flowing through the circuit.

In our exercise, we have a circuit with a 15.0$\mathrm{\Omega }$ resistor connected to a 25.0-V battery. First, we calculate the true current$I_{true}$ as:$$I_{true}=\frac{V}{R}=\frac{25.0}{15.0}=1.67\text{ A}$$Now, placing the ammeter with an effective resistance$R_{amm}=0.9615\mathrm{\Omega }$, we re-calculate the measured current because the ammeter itself adds a small resistance, which slightly alters the total circuit resistance, now seen as:$$I_{measured}=\frac{V}{R_{15}+R_{amm}}=\frac{25.0}{15.0+0.9615}=\text{(value closer but less than 1.67)}$$
This small change helps highlight the importance of accurate ammeter design to minimize measurement errors, which can be further expressed as a percentage error by comparing$I_{measured}$ and$I_{true}$.