Question

14. Judge the equivalent resistance when the following are connected in parallel :
(a) 1ohm and 10^6ohm
(6) 1 ohm and 10^3 ohm and 10^6 ohm​

Answer 1

Given :

(a) Two resistor are connected in parallel

• R₁ = 1 ohm
• R₂ = 10⁶ ohm

(b) Three resistors are connected in parallel

• R₁ = 1 ohm
• R₂ = 10³ ohms
• R₃ = 10⁶ ohms

To find :

The equivalent resistance of the connection.

Solution :

» The reciprocal of the combined resistance of a number of resistance connected in parallel is equal to the sum of the reciprocal of all the individual resistances. .i.e.,

$\dashrightarrow \sf \dfrac{1}{R _{eq} } =\dfrac{1}{R_1} + \dfrac{1}{R_2}$

By substituting the values in the formula,

$\dashrightarrow \sf \dfrac{1}{R _{eq} } =\dfrac{1}{R_1} + \dfrac{1}{R_2}$

$\dashrightarrow \sf \dfrac{1}{R _{eq} } =\dfrac{1}{1} + \dfrac{1}{ {10}^{6} }$

$\dashrightarrow \sf \dfrac{1}{R _{eq} } = \dfrac{ {10}^{6} + 1}{ {10}^{6} }$

$\dashrightarrow \sf \dfrac{1}{R _{eq} } =\dfrac{1000001}{1000000}$

$\dashrightarrow \sf R _{eq} =\dfrac{1000000}{1000001}$

$\dashrightarrow \sf R _{eq}=0.999999 \: ohms$

$\dashrightarrow \sf R _{eq}=1 \: \: ohm \: (approx)$

similarly,

(b) By substituting R₁ = 1 Ω, R₂ = 10³Ω and R₃ = 10⁶

$\dashrightarrow \sf \dfrac{1}{R _{eq} } =\dfrac{1}{R_1} + \dfrac{1}{R_2} + \dfrac{1}{R_3}$

$\dashrightarrow \sf \dfrac{1}{R _{eq} } =\dfrac{1}{1} + \dfrac{1}{ {10}^{3} } + \dfrac{1}{ {10}^{6} }$

$\dashrightarrow \sf \dfrac{1}{R _{eq} } =\dfrac{ {10}^{6} + {10}^{2} + 1}{ {10}^{6} }$

$\dashrightarrow \sf \dfrac{1}{R _{eq} } =\dfrac{ 1000000 + 100 + 1 }{ 1000000}$

$\dashrightarrow \sf \dfrac{1}{R _{eq} } =\dfrac{ 1000101 }{ 1000000}$

$\dashrightarrow \sf R _{eq} =\dfrac{ 1000000}{ 1000101}$

$\dashrightarrow \sf R _{eq} =1.0001 \: ohms$

$\dashrightarrow \sf R _{eq} =1\: ohm$