Chapter Questions
An ideal ammeter would have zero resistance, and an ideal voltmeter would have an infinite resistance. Explain why we would desire these ideal cases when using the meters.
If, in general, $R$ were calculated as $R=V / I$, which circuit arrangement in Part $\mathrm{A}$ of the experiment would have the smallest error? Explain.
(a) Prove that the true resistance $R$ is given by$$R=R^{\prime}\left(1-\frac{R_{\mathrm{a}}}{R^{\prime}}\right)$$where $R^{\prime}=V / I$ is the measured resistance as given by the voltmeter and ammeter readings for measurements done by the arrangement in Fig. $24.2$ or Fig. $24.5 \mathrm{~b}$. Is the true resistance larger or smaller than the apparent resistance?(b) Prove that the true resistance $R$ is given approximately by$$R=R^{\prime}\left(1+\frac{R^{\prime}}{R_{\mathrm{v}}}\right)$$where $R^{\prime}=V / I$ is the measured resistance as given by the voltmeter and ammeter readings for measurements done by the arrangement in Fig. $24.1$ or Fig. $24.5 \mathrm{a}$. (Hint: Use the binomial theorem.)$$\frac{1}{1-\frac{R^{r}}{R_{\mathrm{y}}}} \simeq 1+\frac{R^{\prime}}{R_{\mathrm{v}}}$$Is the true resistance larger or smaller than the apparent resistance? Explain.
For each of the circuits used in the preceding question, for what values of $R$ (large or small) does the error in taking $R$ as equal to $V / I$ become large enough to be important?
Why should the wires connecting the resistances and the bridge be as short as possible?
Suppose that the slide-wire on the bridge did not have a uniform cross section. How would this affect your measurements? Was there any experimental evidence of this?