Question

6).If a Circuit contains resistor of resistance 5 Q and 4 Q connected in series,find the effective resistance of the circuit. (2)​

Answer 1

The effective resistance of the circuit is 9Ω.

Explanation:

In the question, it's been stated that two resistors with resistance 5Ω and 4Ω are connected in series in a circuit. We've been asked to calculate the effective resistance of the circuit.

Well, effective resistance is nothing but the measure of total resistance in a circuit. It's given by:

For series -

$\qquad \quad \boxed{\sf{ R_{e} = R_{1} + R_{2} + R_{3} + ... + R_{n}}}$

For parallel -

$\qquad\:\:\boxed{ \sf{ \dfrac{1}{R_{e}} = \dfrac{1}{R_{1}} + \dfrac{1}{R_{2}} + \dfrac{1}{R_{3}} + ... + \dfrac{1}{R_{n}} }}$

Since our questions is about resistors connected in series, we will be using the first formula and calculating the effective resistance of the circuit.

• Resistor₍₁₎ ⟷ (R₁) = 5Ω
• Resistor₍₂₎ ⟷ (R₂) = 4Ω

Substituting the values in the formula:

$\twoheadrightarrow \quad\sf{ R_{e} = R_{1} + R_{2} + R_{3} + ... + R_{n}}$

$\twoheadrightarrow \quad\sf{ R_{e} = 5 \Omega + 4 \Omega}$

$\twoheadrightarrow \quad \underline{ \boxed{\bf \red{ R_{e} = 9 \Omega}}}$

∴ The effective resistance of the circuit is 9Ω.

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Suppose, if these resistors are connected in a parallel manner, then the effective resistance will be:

$\twoheadrightarrow\quad\sf{\dfrac{1}{R_{e}} = \dfrac{1}{R_{1}} + \dfrac{1}{R_{2}} + \dfrac{1}{R_{3}} + ... + \dfrac{1}{R_{n}}}$

$\twoheadrightarrow\quad\sf{\dfrac{1}{R_{e}} = \dfrac{1}{5} + \dfrac{1}{4}}$

$\twoheadrightarrow\quad\sf{\dfrac{1}{R_{e}} =\bigg( \dfrac{1}{5}\times \dfrac{4}{4} \bigg) + \bigg( \dfrac{1}{4}\times \dfrac{5}{5}\bigg)}$

$\twoheadrightarrow\quad\sf{\dfrac{1}{R_{e}} = \dfrac{4}{20} + \dfrac{5}{20}}$

$\twoheadrightarrow\quad\sf{\dfrac{1}{R_{e}} = \dfrac{4+5}{20}}$

$\twoheadrightarrow\quad\sf{\dfrac{1}{R_{e}} = \dfrac{9}{20}}$

$\twoheadrightarrow\quad\sf{R_{e}= \dfrac{20}{9}}$

$\twoheadrightarrow\quad\sf{R_{e}\approx 2.22 \Omega}$

★ The effective resistance of a circuit containing resistances in a parallel connection is always smaller than any of the individual resistances.