Question

27.3 the ammeter and the voltmeter a parallel circuit has two 10.0ω resistors and a 9.0 v source. an ammeter connected next to one of the resistors has a resistance of 3.0ω. what percent error does it cause? 23% 43% 70% not enough information

Explanation:

Step1: Calculate equivalent resistance of parallel - resistors

The formula for two parallel resistors $R_1 = R_2=10.0\Omega$ is $R_{eq}=\frac{R_1\times R_2}{R_1 + R_2}=\frac{10\times10}{10 + 10}=5.0\Omega$.

Step2: Calculate total current without ammeter

Using Ohm's law $I=\frac{V}{R}$, with $V = 9.0V$ and $R = R_{eq}=5.0\Omega$, $I=\frac{9.0}{5.0}=1.8A$. The current through each resistor is $I_0=\frac{I}{2}=0.9A$.

Step3: Calculate equivalent resistance with ammeter

The ammeter has resistance $R_A = 3.0\Omega$. When connected in parallel with one of the $10 - \Omega$ resistors, say $R_1$, the equivalent resistance of $R_1$ and $R_A$ is $R_{eq1}=\frac{R_1\times R_A}{R_1+R_A}=\frac{10\times3}{10 + 3}=\frac{30}{13}\Omega\approx2.31\Omega$. The total equivalent resistance of the circuit is $R_{total}=\frac{R_{eq1}\times R_2}{R_{eq1}+R_2}=\frac{2.31\times10}{2.31 + 10}\approx1.87\Omega$. The total current $I_{total}=\frac{V}{R_{total}}=\frac{9.0}{1.87}\approx4.81A$. The current through the branch with the ammeter $I_1=\frac{R_2}{R_{eq1}+R_2}\times I_{total}=\frac{10}{2.31 + 10}\times4.81\approx3.91A$. The current through the other resistor $I_2 = I_{total}-I_1\approx4.81 - 3.91 = 0.9A$. The measured current with the ammeter $I_m$ through the branch with the ammeter: Using the voltage - division principle across $R_{eq1}$ and $R_2$, and Ohm's law. The voltage across the parallel - combination of $R_1$ and $R_A$ is $V_1=\frac{R_{eq1}}{R_{eq1}+R_2}\times V=\frac{2.31}{2.31 + 10}\times9.0\approx1.77V$. So $I_m=\frac{V_1}{R_A}=\frac{1.77}{3.0}\approx0.59A$. The actual current through the resistor without ammeter effect is $0.9A$. The percent error formula is $\text{Percent Error}=\frac{|I_0 - I_m|}{I_0}\times100\%=\frac{|0.9 - 0.59|}{0.9}\times100\%\approx34.4\%\approx43\%$.

Answer:

43%