Question

Unlike the idealized ammeter described in Section 25.4, any real ammeter has a nonzero resistance. (a) An ammeter with resistance \(R_A\) is connected in series with a resistor \(R\) and a battery of emf \(\varepsilon\) and internal resistance r. The current measured by the ammeter is \(I_A\). Find the current through the circuit if the ammeter is removed so that the battery and the resistor form a complete circuit. Express your answer in terms of \(I_A\), \(r\), \(R_A\), and \(R\). The more "ideal" the ammeter, the smaller the difference between this current and the current IA. (b) If \(R\) = 3.80 \(\Omega\), \(\varepsilon\) = 7.50 V, and \(r\) = 0.45 \(\Omega\), find the maximum value of the ammeter resistance \(R_A\) so that \(I_A\) is within 1.0% of the current in the circuit when the ammeter is absent. (c) Explain why your answer in part (b) represents a \(maximum\) value.

Short answer

(a) \[ I = \frac{I_A (R + R_A + r)}{R + r} \]. (b) Maximum \( R_A \) is 0.0425 \( \Omega \). (c) Higher \( R_A \) increases the error in the current measurement, exceeding 1% difference.

Step-by-step solution

Express Current with Ammeter

When the ammeter is connected, the current through the circuit is measured as \( I_A \). We express this current using Ohm's Law, taking into account all resistances in the series:\[I_A = \frac{\varepsilon}{R + R_A + r}\]

Find Current without Ammeter

If the ammeter is removed, the total resistance in the circuit decreases to just \( R + r \). The new current \( I \) can be found using Ohm's Law:\[I = \frac{\varepsilon}{R + r}\]

Relate Current without Ammeter to Measured Current

We aim to express \( I \) in terms of the measured current \( I_A \). Starting with the expression for \( I \), we can substitute \( \varepsilon \) from the expression for \( I_A \):\[\varepsilon = I_A \cdot (R + R_A + r)\]Then substitute this into the equation for \( I \):\[I = \frac{I_A (R + R_A + r)}{R + r}\]

Solve for Maximum Ammeter Resistance

For \( I_A \) to be within 1% of \( I \), we set up the inequality:\[I_A \geq 0.99 \times I\]Substitute the expression from Step 3:\[I_A \geq 0.99 \times \frac{I_A (R + R_A + r)}{R + r}\]Simplify this inequality to find the condition on \( R_A \):\[0.99 (R + r) \geq R + R_A + r - R - r\]Solve for \( R_A \) with given values of \( R = 3.80 \, \Omega \), \( r = 0.45 \, \Omega \):\[0.99(4.25) \geq 3.80 + R_A + 0.45 - 4.25 = R_A \]\[R_A \leq 0.0425\, \Omega\]The maximum resistance \( R_A \) is approximately 0.0425 \( \Omega \).

Justify Maximum Value for Ammeter Resistance

The reason the derived \( R_A \) is a maximum value is because any higher resistance would increase the discrepancy between \( I_A \) and \( I \), exceeding the 1% difference allowed. Increasing \( R_A \) further would result in more significant resistance in the circuit, thus lowering the current \( I_A \) compared to \( I \), and violating the condition.

Key concepts

Ohm's Law

Ohm's Law is a fundamental principle in electronics that relates the voltage (V), current (I), and resistance (R) in an electrical circuit. It is expressed by the formula:\[ V = I \cdot R \]This equation shows that the voltage across a resistor in a circuit is equal to the product of the current flowing through it and the resistance. Ohm's Law is crucial because it helps determine how current flows in a circuit under different conditions. It's a simple yet powerful way to understand the behaviour of electrical components.
Applicable in most scenarios, Ohm's Law helps find unknown values like current when the voltage and resistance are known, or vice versa. In this exercise, Ohm's Law is applied to analyze the difference in current with and without the ammeter, illustrating how changes in resistance (due to the ammeter) affect overall current.

Circuit Analysis

Circuit analysis is the process of finding the currents and voltages in every component of an electrical circuit. It involves using laws and theorems, like Ohm's Law and Kirchhoff's laws, to predict how electrical components will function together.

• Understand the components: Resistors, batteries, and ammeters.
• Consider total resistance: Add up all resistances in series.
• Apply laws: Use Ohm's Law or other rules to find the desired quantities.

In the given problem, circuit analysis helps compare the current with the ammeter in place and the current when it is removed. By understanding circuit resistance and how it changes, we can predict the circuit's behaviour accurately.

Current Measurement

Measuring current accurately is vital for diagnosing and analyzing circuits. Current is typically measured in amperes (A) and represents the flow of electric charge.
An ammeter is a device specifically designed to measure this flow and is usually connected in series with the circuit's components. This is essential because it allows the current to pass through the ammeter, giving an accurate reading of the current flowing through the entire circuit.
To ensure minimal interference, an ideal ammeter should have zero resistance. However, real ammeters do have some resistance which can affect the measurement slightly. This resistance must be minimized to ensure accurate readings. Understanding how this small resistance affects current measurement is crucial, as demonstrated in the exercise when finding an acceptable value for ammeter resistance to keep measurement errors within 1%.

Ideal vs Real Ammeter

An "Ideal Ammeter" is a theoretical concept where the ammeter has zero resistance. This means it does not affect the circuit's operation. Such an ammeter does not alter the current it measures, providing a perfectly accurate reading. In practical terms, this ideal doesn't exist, but it's a useful concept for understanding fundamental electronics.
In contrast, a "Real Ammeter" has some non-zero internal resistance. This resistance affects the overall resistance in a circuit, slightly altering the current. When inserting a real ammeter, the measured current is slightly less than what it would be without the ammeter.
Understanding the difference is crucial for anyone working with circuits as it helps to account for and minimize errors in measurement. In our problem, we derived a maximum permissible ammeter resistance to ensure the real ammeter's effect on current measurement stayed within a 1% tolerance. This highlights the balance between practical measurement and theoretical ideals.