Question

An ammeter with an internal resistance of \(53 \Omega\) measures a current of \(5.25 \mathrm{~mA}\) in a circuit containing a battery and a total resistance of \(1130 \Omega\). The insertion of the ammeter alters the resistance of the circuit, and thus the measurement does not give the actual value of the current in the circuit without the ammeter. Determine the actual value of the current.

Short answer

Answer: The actual value of the current in the circuit without the ammeter is approximately 5.44 mA.

Step-by-step solution

Calculate the original resistance of the circuit without the ammeter

Without the ammeter, the total resistance of the circuit is \(R = 1130 \Omega\). We want to find the actual current without the ammeter.

Calculate the equivalent resistance of the circuit with the ammeter

The internal resistance of the ammeter is \(R_A = 53 \Omega\). When the ammeter is inserted into the circuit, the equivalent resistance of the circuit becomes the sum of the original resistance and the ammeter's resistance: \(R_{eq} = R + R_A\).

Calculate the actual current in the circuit

The ammeter measures a current of \(5.25 \mathrm{mA}\) in the circuit with an altered resistance of \(R_{eq}\). Let \(I_{measured}\) be the measured current and \(I_{actual}\) be the actual current we want to find. We can apply Ohm's Law (V=IR) to both the circuit with and without the ammeter: \(I_{actual} R = I_{measured} R_{eq}\) Now, we need to solve for \(I_{actual}\). Since we know the value of the resistances and the measured current, we can plug in values: \(I_{actual} = \frac{I_{measured} R_{eq}}{R}\)

Plug in values and solve for the actual current

We can now plug in the values for the measured current and resistances into the formula: \(I_{actual} = \frac{(5.25 \mathrm{~mA})(1130 \Omega + 53 \Omega)}{1130 \Omega}\) \(I_{actual} = \frac{5.25 \mathrm{~mA} \cdot 1183 \Omega}{1130 \Omega}\) \(I_{actual} \approx 5.44 \mathrm{~mA}\) Therefore, the actual value of the current in the circuit without the ammeter is approximately \(5.44 \mathrm{~mA}\).

Key concepts

Ammeter

An ammeter is a crucial instrument used in electrical circuits to measure the current flowing through the circuit. It is connected in series to the circuit, meaning it becomes a part of the circuit's electric path. This positioning allows it to directly measure the current. However, every ammeter has an internal resistance, which can affect the circuit’s total resistance when it is inserted.

The internal resistance of an ammeter can cause a slight deviation in the actual current measurement. For example, in the given problem, the ammeter’s internal resistance was 53 Ohms. This resistance affected the circuit's total resistance and, consequently, the measured current. To ensure accurate readings, it's crucial to consider an ammeter's internal resistance in some precise measurements or calculations.

Resistance

Resistance is the opposition that a material offers to the flow of electric current. It is measured in ohms (Ω). In circuits, resistance affects how much current will flow for a given voltage, as explained by Ohm's Law (V=IR).

In the problem at hand, the circuit had a given total resistance of 1130 Ω. When the ammeter, with its internal resistance of 53 Ω, was added to the circuit, the total resistance increased. This new total resistance had to be considered to find the accurate current flowing through the circuit.

• Internal Resistance: Causes the total resistance to increase when an ammeter is added.
• Circuit Resistance: A sum of all resistive components in a given path.

Understanding resistance is essential for analyzing circuits and predicting how they will behave under various conditions.

Measured Current

Measured current is the value that an ammeter displays when it is part of a circuit. This current can differ from the actual current due to the added resistance from the ammeter. In the mentioned problem, the measured current was 5.25 mA.

Due to the additional resistance introduced by the ammeter, the measured current was less than the actual current in the circuit without the ammeter. To determine the actual current, we first find the total resistance with the ammeter and then use Ohm's Law to correct the measurement. It's important to differentiate between measured current and actual current to ensure electrical devices function efficiently and safely.

Circuit Analysis

Circuit analysis is the process of determining the currents and voltages in a circuit. This includes understanding how different components and their characteristics, like resistance, impact the overall circuit behavior.

There are several methods to analyze circuits. Ohm’s Law is one fundamental tool used in these analyses. In the given problem, we used Ohm's Law to compare the circuit's behavior with and without the ammeter. This involved knowing the equivalence of circuit resistances and using these values to calculate the actual current.

• Ohm's Law Application: Formulating equations to relate currents with resistances and voltages.
• Series and Parallel Configurations: Affect how components influence total resistance.

Circuit analysis ensures we can calculate accurately and predictively manage components in an electrical circuit to work efficiently.