Respuesta :
The formula to calculate the total resistance of a parallel circuit is:
[tex]\frac{1}{R_T}=\frac{1}{R_1}+\frac{1}{R_2}+...+\frac{1}{R_n}[/tex]
Where [tex]R_1, R_2, ... R_n[/tex] are the individual resistances
Let's now apply the formula to each circuit:
A. [tex]6.67 \Omega[/tex]
A parallel circuit with a 20-ohm resistor and a 10-ohm resistor.
The equivalent resistance is:
[tex]\frac{1}{R_T}=\frac{1}{20 \Omega}+\frac{1}{10 \Omega}=\frac{3}{20 \Omega}[/tex]
[tex]R_T = \frac{20}{3}\Omega =6.67 \Omega[/tex]
B. [tex]5.0 \Omega[/tex]
A parallel circuit with two 20-ohm resistors and a 10-ohm resistor
The equivalent resistance is:
[tex]\frac{1}{R_T}=\frac{1}{20\Omega}+\frac{1}{20 \Omega}+\frac{1}{10 \Omega}=\frac{4}{20 \Omega}[/tex]
[tex]R_T = \frac{20}{4}\Omega =5.0 \Omega[/tex]
C. [tex]8.57 \Omega[/tex]
A parallel circuit with a 15-ohm light bulb and a 20-ohm resistor.
The equivalent resistance is:
[tex]\frac{1}{R_T}=\frac{1}{20 \Omega}+\frac{1}{15 \Omega}=0.1167 \Omega ^{-1}[/tex]
[tex]R_T = \frac{1}{0.1167}\Omega =8.57 \Omega[/tex]
D. [tex]14.3 \Omega[/tex]
A parallel circuit with two 100-ohm resistors and a 20-ohm resistor.
The equivalent resistance is:
[tex]\frac{1}{R_T}=\frac{1}{20\Omega}+\frac{1}{100 \Omega}+\frac{1}{100 \Omega}=\frac{7}{100 \Omega}[/tex]
[tex]R_T = \frac{100}{7}\Omega =14.3 \Omega[/tex]
E. [tex]6.1 \Omega[/tex]
A parallel circuit with a 10-ohm, 20-ohm, 100-ohm and 200-ohm resistor.
The equivalent resistance is:
[tex]\frac{1}{R_T}=\frac{1}{10\Omega}+\frac{1}{20 \Omega}+\frac{1}{100 \Omega}+\frac{1}{200 \Omega}=\frac{33}{200 \Omega}[/tex]
[tex]R_T = \frac{200}{33}\Omega =6.1 \Omega[/tex]
(A) The total resistance for parallel circuit with a 20-ohm resistor and a 10-ohm resistor is 6.67 ohms.
(B) The total resistance for parallel circuit with two 20-ohm resistor and a 10-ohm resistor is 5 ohms.
(C) The total resistance for parallel circuit with a 15-ohm light bulb and a 20-ohm resistor is 8.57 ohms.
(D) The total resistance for parallel circuit with a 10-ohm, a 20-ohm, 100-ohm and 200-ohm resistor is 6.06 ohms.
(E) The total resistance for parallel circuit with a 10-ohm, a 20-ohm, 100-ohm and 200-ohm resistor is 6.06 ohms.
The total resistance for the parallel combination of resistors is given as
[tex]\dfrac{1}{R_{total}}=\dfrac{1}{R_{1}}+\dfrac{1}{R_{2}}+...........................+\dfrac{1}{R_{n}}[/tex]
And the total resistance for the series combination of resistors is,
[tex]R_{total}=R_{1}+R_{2}+................................+R_{n}[/tex]
(A)
For parallel circuit of 20-ohm and 10-ohm resistors, the total resistance is,
[tex]\dfrac{1}{R_{total}}=\dfrac{1}{R_{1}}+\dfrac{1}{R_{2}}\\\dfrac{1}{R_{total}}=\dfrac{1}{20}+\dfrac{1}{10}\\R_{total}=\dfrac{20 \times 10}{20+10}\\R_{total}=6.67 \;\rm ohms[/tex]
Thus, the total resistance for parallel circuit with a 20-ohm resistor and a 10-ohm resistor is 6.67 ohms.
(B)
For parallel circuit with two 20-ohm and a 10-ohm resistor, the total resistance is,
[tex]\dfrac{1}{R_{total}}=\dfrac{1}{R_{1}}+\dfrac{1}{R_{2}}+\dfrac{1}{R_{3}}\\\dfrac{1}{R_{total}}=\dfrac{1}{20}+\dfrac{1}{20}+\dfrac{1}{10}\\R_{total}=\dfrac{20}{4}\\R_{total}=5 \;\rm ohms[/tex]
Thus, the total resistance for parallel circuit with two 20-ohm resistor and a 10-ohm resistor is 5 ohms.
(C)
For parallel circuit with a 15-ohm light bulb and a 20-ohm resistor, the total resistance is,
[tex]\dfrac{1}{R_{total}}=\dfrac{1}{R_{1}}+\dfrac{1}{R_{2}}\\\dfrac{1}{R_{total}}=\dfrac{1}{15}+\dfrac{1}{20}\\R_{total}=\dfrac{15 \times 20}{15+20}\\R_{total}=8.57 \;\rm ohms[/tex]
Thus, the total resistance for parallel circuit with a 15-ohm light bulb and a 20-ohm resistor is 8.57 ohms.
(D)
For parallel circuit with two 100-ohm and a 20-ohm resistor, the total resistance is,
[tex]\dfrac{1}{R_{total}}=\dfrac{1}{R_{1}}+\dfrac{1}{R_{2}}+\dfrac{1}{R_{3}}\\\dfrac{1}{R_{total}}=\dfrac{1}{100}+\dfrac{1}{100}+\dfrac{1}{20}\\R_{total}=\dfrac{100}{7}\\R_{total}=14.28 \;\rm ohms[/tex]
Thus, the total resistance for parallel circuit with two 100-ohm and a 20-ohm resistor is 14.28 ohms.
(E)
For parallel circuit with a 10-ohm, a 20-ohm, 100-ohm and 200-ohm resistor, the total resistance is,
[tex]\dfrac{1}{R_{total}}=\dfrac{1}{R_{1}}+\dfrac{1}{R_{2}}+\dfrac{1}{R_{3}}+\dfrac{1}{R_{4}}\\\dfrac{1}{R_{total}}=\dfrac{1}{10}+\dfrac{1}{20}+\dfrac{1}{100}+\dfrac{1}{200}\\R_{total}=\dfrac{200}{33}\\R_{total}=6.06 \;\rm ohms[/tex]
Thus, the total resistance for parallel circuit with a 10-ohm, a 20-ohm, 100-ohm and 200-ohm resistor is 6.06 ohms.
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