The absolute error and % error in the measurement of equivalent resistance are :
Series → 7 Ohm and 2%
Parallel → 5 Ohm and 7%
Further explanation
Electrical circuits can generally be divided into two types , i.e :
1. Series Circuit
In series circuit, the electric current flowing on each resistor is always the same as the total current.
To find the total resistances you can use the following formula:
[tex]\large {\boxed {R_s = R_1 + R_2 + R_3 + ...} }[/tex]
2. Parallel Circuit
In parallel circuits, the electrical voltage at each resistor is always the same as the source voltage.
To find the total resistances you can use the following formula:
[tex]\large {\boxed {\frac{1}{R_p} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + ...} }[/tex]
Let's tackle the problem now !
Given:
[tex]R_1 = 100 \pm 3 ~ Ohm[/tex]
[tex]R_2 = 200 \pm 4 ~ Ohm[/tex]
Unknown:
Absolute Error = ?
% Error = ?
Solution:
In Series Circuit :
[tex]R_s = R_1 + R_2[/tex]
[tex]R_s = ( 100 \pm 3 ) + ( 200 \pm 4 )[/tex]
[tex]\large {\boxed {R_s = 300 \pm 7 ~ Ohm} }[/tex]
Absolute Error = 7 Ohm
% Error = (7/300) x 100% ≈ 2%
In Parallel Circuit :
[tex]\frac{1}{R_p} = \frac{1}{R_1} + \frac{1}{R_2}[/tex]
[tex]\frac{1}{R_p} = \frac{R_2}{R_1 ~ R_2} + \frac{R_1}{R_2 ~ R_1}[/tex]
[tex]\frac{1}{R_p} = \frac{R_1 + R_2}{R_1 ~ R_2}[/tex]
[tex]R_p = \frac{R_1 ~ R_2}{R_1 + R_2}[/tex]
[tex]R_p = \frac{100 \times 200}{100 + 200}[/tex]
[tex]R_p = \frac{20000}{300}[/tex]
[tex]R_p = \frac{200}{3} ~ Ohm[/tex]
[tex]R_p \approx 67 ~ Ohm[/tex]
To find the % Error , we will use this following formula :
[tex]\frac{\Delta R_p}{R_p} = \frac{\Delta R_1}{R_1} + \frac{\Delta R_2}{R_2} + \frac{\Delta (R_1 +R_2)}{R_1 + R_2}[/tex]
[tex]\frac{\Delta R_p}{R_p} = \frac{3}{100} + \frac{4}{200} + \frac{7}{300}[/tex]
[tex]\frac{\Delta R_p}{R_p} = \frac{11}{150}[/tex]
[tex]\frac{\Delta R_p}{R_p} = \frac{11}{150} \times 100{\%}[/tex]
[tex]\frac{\Delta R_p}{R_p} \approx 7{\%}[/tex]
[tex]Absolute ~ Error = \Delta R_p[/tex]
[tex]Absolute ~ Error = \frac{11}{150} \times \frac{200}{3}[/tex]
[tex]Absolute ~ Error = \frac{44}{9}[/tex]
[tex]Absolute ~ Error \approx 5 ~ Ohm[/tex]
[tex]\large {\boxed {R_p = 67 \pm 5 ~ Ohm} }[/tex]
Learn more
- The three resistors : https://brainly.com/question/9503202
- A series circuit : https://brainly.com/question/1518810
- Compare and contrast a series and parallel circuit : https://brainly.com/question/539204
Answer details
Grade: High School
Subject: Physics
Chapter: Current of Electricity
Keywords: Series , Parallel , Measurement , Absolute , Error , Combination , Resistor , Resistance , Ohm
