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title: "To determine an unknown resistance \\(R\\), a student considers two circuit configurations using a nonideal voltmeter with internal resistance \\(R_V\\) and a nonideal ammeter with internal resistance \\(R_A\\).  Scheme 1: The voltmeter is connected in parallel across \\(R\\), and the ammeter is connected in series with the power source to measure the total current entering the parallel combination. Scheme 2: The ammeter is connected in series with \\(R\\), and the voltmeter is connected in parallel across the series combination of \\(R\\) and the ammeter.  For each scheme, the measured resistance is defined as \\(R_{\\text{meas}} = \\dfrac{V}{I}\\), where \\(V\\) is the voltmeter reading and \\(I\\) is the ammeter reading. Which scheme yields a measured resistance that is less than the true resistance \\(R\\), and what is the exact fractional error \\(\\dfrac{R_{\\text{meas}} – R}{R}\\) for that scheme?  | Choice | Scheme with \\(R_{\\text{meas}} < R\\) | Fractional error \\(\\dfrac{R_{\\text{meas}} – R}{R}\\) | |—|—|—| | A | Scheme 1 | \\(\\dfrac{R_A}{R}\\) | | B | Scheme 1 | \\(-\\dfrac{R}{R + R_V}\\) | | C | Scheme 2 | \\(-\\dfrac{R}{R + R_V}\\) | | D | Scheme 2 | \\(\\dfrac{R_A}{R + R_A}\\) |"
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url: "https://nerd-notes.com/ubq/118474/"
date_modified: "2026-08-04T08:10:33+00:00"
---

# To determine an unknown resistance \(R\), a student considers two circuit configurations using a nonideal voltmeter with internal resistance \(R_V\) and a nonideal ammeter with internal resistance \(R_A\).

Scheme 1: The voltmeter is connected in parallel across \(R\), and the ammeter is connected in series with the power source to measure the total current entering the parallel combination.
Scheme 2: The ammeter is connected in series with \(R\), and the voltmeter is connected in parallel across the series combination of \(R\) and the ammeter.

For each scheme, the measured resistance is defined as \(R_{\text{meas}} = \dfrac{V}{I}\), where \(V\) is the voltmeter reading and \(I\) is the ammeter reading. Which scheme yields a measured resistance that is less than the true resistance \(R\), and what is the exact fractional error \(\dfrac{R_{\text{meas}} – R}{R}\) for that scheme?

| Choice | Scheme with \(R_{\text{meas}} < R\) | Fractional error \(\dfrac{R_{\text{meas}} – R}{R}\) |
|—|—|—|
| A | Scheme 1 | \(\dfrac{R_A}{R}\) |
| B | Scheme 1 | \(-\dfrac{R}{R + R_V}\) |
| C | Scheme 2 | \(-\dfrac{R}{R + R_V}\) |
| D | Scheme 2 | \(\dfrac{R_A}{R + R_A}\) |

To determine an unknown resistance \(R\), a student considers two circuit configurations using a nonideal voltmeter with internal resistance \(R_V\) and a nonideal ammeter with internal resistance \(R_A\).

Scheme 1: The voltmeter is connected in parallel across \(R\), and the ammeter is connected in series with the power source to measure the total current entering the parallel combination.
Scheme 2: The ammeter is connected in series with \(R\), and the voltmeter is connected in parallel across the series combination of \(R\) and the ammeter.

For each scheme, the measured resistance is defined as \(R_{\text{meas}} = \dfrac{V}{I}\), where \(V\) is the voltmeter reading and \(I\) is the ammeter reading. Which scheme yields a measured resistance that is less than the true resistance \(R\), and what is the exact fractional error \(\dfrac{R_{\text{meas}} - R}{R}\) for that scheme?

| Choice | Scheme with \(R_{\text{meas}} < R\) | Fractional error \(\dfrac{R_{\text{meas}} - R}{R}\) |
|---|---|---|
| A | Scheme 1 | \(\dfrac{R_A}{R}\) |
| B | Scheme 1 | \(-\dfrac{R}{R + R_V}\) |
| C | Scheme 2 | \(-\dfrac{R}{R + R_V}\) |
| D | Scheme 2 | \(\dfrac{R_A}{R + R_A}\) |

![Two circuit schematics side-by-side labeled Scheme 1 on the left and Scheme 2 on the right. In Scheme 1, a DC voltage source on the left side of a rectangular loop connects in series to a circle labeled A at the top. The top wire splits into two parallel vertical branches: the left branch contains a circle labeled V, and the right branch contains a zigzag resistor symbol labeled R. The branches rejoin at the bottom wire back to the voltage source. In Scheme 2, a DC voltage source on the left side connects to top and bottom wires. A circle labeled V is in a vertical branch across the main wires. To the right of V, the top wire connects in series to a circle labeled A, followed by a zigzag resistor labeled R in a vertical branch returning to the bottom wire. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785831033-KS4ZuR.jpg)

- **A.** Scheme 1; \(\dfrac{R_A}{R}\)
- **B.** Scheme 1; \(-\dfrac{R}{R + R_V}\)
- **C.** Scheme 2; \(-\dfrac{R}{R + R_V}\)
- **D.** Scheme 2; \(\dfrac{R_A}{R + R_A}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118474/*