Question

The ammeter your physics instructor uses for in-class demonstrations has internal resistance \(R_{\mathrm{i}}=75 \Omega\) and measures a maximum current of \(1.5 \mathrm{~mA}\). The same ammeter can be used to measure currents of much greater magnitudes by wiring a shunt resistor of relatively small resistance, \(R_{\text {shunt }}\), in parallel with the ammeter. (a) Sketch the circuit diagram, and explain why the shunt resistor connected in parallel with the ammeter allows it to measure larger currents. (b) Calculate the resistance the shunt resistor has to have to allow the ammeter to measure a maximum current of 15 A.

Short answer

Question: Sketch a circuit diagram with an ammeter and a shunt resistor connected in parallel and explain how this setup allows the ammeter to measure larger currents. Calculate the resistance of the shunt resistor that must be used to measure a maximum current of 15 A, given that the ammeter has an internal resistance of 75 Ohms and a maximum current of 1.5 mA. Answer: When an ammeter and a shunt resistor are connected in parallel, the current gets divided between them based on their respective resistances. The ammeter measures the current flowing through it and, by knowing the resistance values of both the ammeter and the shunt resistor, we can calculate the total current flowing through the circuit. To measure a maximum current of 15 A, a shunt resistor with a resistance of approximately 0.0075 Ohms is required.

Step-by-step solution

Sketch the circuit diagram with the ammeter and shunt resistor connected in parallel

First, draw a simple circuit with a battery, a resistor representing the device we want to measure the current of (load), and the ammeter connected in series. Next, add the shunt resistor in parallel with the ammeter, which means connecting one end of the shunt resistor to the same point where the ammeter connects to the circuit, and the other end to the same point where the ammeter connects to the load. This creates a parallel connection between the ammeter and the shunt resistor.

Understand the parallel connection of the ammeter and shunt resistor

When the ammeter and shunt resistor are connected in parallel, the current flowing through the circuit is divided between the two resistors according to their resistance. The ammeter measures the current flowing through it, and since we know the resistance of the ammeter and the shunt resistor, we can determine the total current flowing through the circuit. In this way, the ammeter can measure larger currents that would otherwise be beyond its maximum limit. The combined resistance of the ammeter and shunt resistor effectively becomes smaller, allowing more current to flow.

Write down equations to determine the shunt resistor's resistance

We know the formula for the equivalent resistance of parallel resistors is: \(\frac{1}{R_{total}} = \frac{1}{R_{i}} + \frac{1}{R_{shunt}}\) We also know the maximum current the ammeter can measure (1.5 mA) and the maximum current we want it to measure (15 A). Let's denote the current passing through the ammeter as I_i and the current passing through the shunt resistor as I_shunt. We can write down the following equations: \(I_{total} = I_{i} + I_{shunt}\) \(I_i = \frac{V}{R_i}\) \(I_{shunt} = \frac{V}{R_{shunt}}\) We want to find the value of R_shunt that enables the ammeter to measure a maximum current of 15 A.

Solving for R_shunt when the maximum current is 15 A

First, we calculate the maximum voltage drop across the internal resistance of the ammeter: \(V_{max} = R_i \times I_{max} = 75 \Omega \times 1.5 \times 10^{-3} A = 0.1125 V\) To measure a total current of 15 A, we need the current through the shunt resistor to be: \(I_{shunt} = I_{total} - I_i = 15 A - 1.5 \times 10^{-3} A = 14.9985 A\) Now we can calculate the required shunt resistance: \(R_{shunt} = \frac{V_{max}}{I_{shunt}} = \frac{0.1125 V}{14.9985 A} = 0.0075 \Omega\) The shunt resistor must have a resistance of approximately 0.0075 Ohms to allow the ammeter to measure a maximum current of 15 A.

Key concepts

Ammeter

An ammeter is an essential instrument for measuring electrical current in a circuit. They are often connected in series with the component under test, allowing the current to pass through the ammeter. This method ensures that the ammeter can measure the current flowing through the circuit directly. Ammeters are designed to have very low internal resistance. This low resistance helps minimize their impact on the overall current flow within the circuit.
Furthermore, ammeters have a maximum current level they can measure, which is dictated by their design specifications. Beyond this limit, they can become damaged. Hence, to measure larger currents, additional components like shunt resistors are used.

Shunt Resistor

A shunt resistor is a special type of resistor used in conjunction with ammeters to expand their range of measurable currents. By having a low resistance value, the shunt resistor is connected in parallel with the ammeter. As a result, it allows most of the current to bypass the ammeter, protecting it from overloading.
The primary role of a shunt resistor is to divert the majority of the current away from the ammeter, ensuring that the ammeter reads only a fraction of the total current. This setup allows the ammeter to safely measure higher currents than it normally would. Shunt resistors are carefully chosen to have the right resistance to share the current appropriately between itself and the ammeter.

Parallel Connection

In the context of circuit analysis, a parallel connection refers to the arrangement where components share the same two nodes, effectively creating multiple paths for the current to travel. When the ammeter and the shunt resistor are placed in parallel, the circuit ensures that the total current is distributed between the two.
The defining feature of parallel connections is that voltage across all components is equal, making it possible to apply Kirchhoff's voltage law efficiently. This law states that the sum of the voltages around any closed loop in a circuit must equal zero. For parallel circuits, the equivalent resistance can be calculated using the formula \[ \frac{1}{R_{total}} = \frac{1}{R_{1}} + \frac{1}{R_{2}} + \dots \] where \( R_{1} \) and \( R_{2} \) are the resistances of the individual components in parallel.

Equivalent Resistance

Equivalent resistance is an essential concept in circuit analysis, particularly when dealing with complex circuits involving parallel and series components. In a parallel circuit, as with the ammeter and shunt resistor, the equivalent resistance decreases, allowing more current to flow through the entire circuit.
For parallel circuits, the equivalent resistance is calculated by \[ \frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \dots \] This formula shows that as more paths for current are created (by adding more resistors in parallel), the total resistance declines. This decrease in resistance is why parallel circuits can handle higher currents safely, since they lower the potential burden on any single pathway, thus protecting sensitive components like the ammeter.