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Physics Chapter 11 : Topic 12
Resistors in Parallel
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When resistors are connected in parallel, the potential difference across each resistor is:
Reasoning: In a parallel combination, all resistors are connected between the same two points, so the voltage across each resistor is equal.
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The formula for combined resistance of resistors in parallel is derived using:
Reasoning: Kirchhoff’s first law gives the relation between branch currents, while Ohm’s law relates current and voltage.
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Two resistors $\mathbf{R}_{\mathbf{1}}$ and $\mathbf{R}_{\mathbf{2}}$ are connected in parallel. The reciprocal of equivalent resistance is:
Reasoning: In parallel circuits, conductances add, giving the reciprocal resistance formula.
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In a parallel circuit, the total current supplied by the battery is:
Reasoning: Kirchhoff’s first law states that current entering a junction equals the sum of currents leaving it.
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A battery of voltage $\mathbf{V}$ is connected to two parallel resistors $\mathbf{R}_{\mathbf{1}}$ and $\mathbf{R}_{\mathbf{2}}$. The branch currents are:
Reasoning: Each branch has the same voltage, so Ohm’s law gives $\mathbf{I = V\ /\ R}$.
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For two resistors connected in parallel, the equivalent resistance is:
Reasoning: Adding another parallel path decreases the total opposition to current flow.
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Two resistors of $\mathbf{6\ \Omega}$ and $\mathbf{3\ \Omega}$ are connected in parallel. Their equivalent resistance is:
Reasoning: $\mathbf{R =}$ $\frac{\mathbf{R}_{\mathbf{1\ }}\mathbf{R}_{\mathbf{2}}}{\mathbf{R}_{\mathbf{1}}\mathbf{\ + \ }\mathbf{R}_{\mathbf{2}}}\mathbf{=}\frac{\mathbf{6\ \times \ 3}}{\mathbf{6\ + \ 3}}\mathbf{= 2\ \Omega}$
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When an additional resistor is connected in parallel, the total resistance:
Reasoning: A new parallel path allows more current to flow, reducing equivalent resistance.
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The equivalent resistance of three resistors $\mathbf{R}_{\mathbf{1}}$, $\mathbf{R}_{\mathbf{2}}$, and $\mathbf{R}_{\mathbf{3}}$ in parallel is given by:
Reasoning: The reciprocals of resistances (conductances) add in a parallel combination.
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In deriving parallel resistance, Kirchhoff’s first law gives:
Reasoning: The total current entering a junction equals the sum of currents leaving through branches.
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In a parallel combination, the branch having the smallest resistance carries:
Reasoning: Since voltage is constant, $\mathbf{I = V\ /\ R}$ shows that smaller resistance allows greater current.
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Three resistors of $\mathbf{4\ \Omega}$, $\mathbf{6\ \Omega}$ and $\mathbf{12\ \Omega}$ are connected in parallel. Their equivalent resistance is:
Reasoning: $\frac{\mathbf{1}}{\mathbf{R}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{4}}\mathbf{+}\frac{\mathbf{1}}{\mathbf{6}}\mathbf{+}\frac{\mathbf{1}}{\mathbf{12}}\mathbf{=}\frac{\mathbf{6}}{\mathbf{12}}$ $\mathbf{\Longrightarrow}$ Therefore, $\mathbf{R = 2\ \Omega}$
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The equivalent resistance formula for two parallel resistors can be written as:
Reasoning: Rearranging the reciprocal formula gives the product-over-sum expression.
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A $\mathbf{12\ V}$ battery is connected across two parallel resistors of $\mathbf{6\ \Omega\ }$and $\mathbf{4\ \Omega}$. The total current is:
Reasoning: Equivalent resistance $\mathbf{\Longrightarrow}$ $\mathbf{R =}$ $\frac{\mathbf{R}_{\mathbf{1}}\mathbf{\ }\mathbf{R}_{\mathbf{2}}}{\mathbf{R}_{\mathbf{1}}\mathbf{\ + \ }\mathbf{R}_{\mathbf{2}}}\mathbf{=}\frac{\mathbf{6\ \times \ 4}}{\mathbf{6\ + \ 4}}\mathbf{= 2.4\ \Omega}$Current $\mathbf{\Longrightarrow}$ $\mathbf{I =}$ $\frac{\mathbf{V}}{\mathbf{R}}\mathbf{=}\frac{\mathbf{12}}{\mathbf{2.4}}\mathbf{= 5\ A}$
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The current in a parallel circuit divides because:
Reasoning: Current splits according to the resistance of each branch.
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If one branch of a parallel circuit is disconnected, then:
Reasoning: Other branches still provide complete paths for current flow.
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Two identical resistors $\mathbf{R}$ are connected in parallel. Their equivalent resistance is:
Reasoning: $\frac{\mathbf{1}}{\mathbf{R}_{\mathbf{eq}}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{R}}\mathbf{+}\frac{\mathbf{1}}{\mathbf{R}}\mathbf{=}\frac{\mathbf{2}}{\mathbf{R}}$ $\mathbf{\Longrightarrow}$ Therefore, $\mathbf{R}_{\mathbf{eq}}\mathbf{=}$ $\frac{\mathbf{R}}{\mathbf{2}}$
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In a parallel circuit, the total power supplied by the source equals:
Reasoning: Conservation of energy requires source power to equal total power consumed.
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The derivation of parallel resistance depends on the fact that:
Reasoning: All branches are connected between the same two points, giving equal potential difference.
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A $\mathbf{10\ \Omega}$ resistor and a $\mathbf{20\ \Omega}$ resistor are connected in parallel. The equivalent resistance is:
Reasoning: $\mathbf{R =}$ $\frac{\mathbf{R}_{\mathbf{1}}\mathbf{\ }\mathbf{R}_{\mathbf{2}}}{\mathbf{R}_{\mathbf{1}}\mathbf{\ + \ }\mathbf{R}_{\mathbf{2}}}\mathbf{=}\frac{\mathbf{10\ \times \ 20}}{\mathbf{10\ + \ 20}}\mathbf{= 6.67\ }\mathbf{\Omega}$
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The current through each branch in a parallel circuit depends on:
Reasoning: Branch current is determined by $\mathbf{I = V\ /\ R}$.
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If one resistor in a parallel circuit decreases in value, the total current:
Reasoning: Lower resistance increases current in that branch, increasing total current.
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A circuit has two parallel resistors with equivalent resistance $\mathbf{5\ \Omega}$. If one resistor is $\mathbf{10\ \Omega}$, the other resistor is:
Reasoning: Two equal resistors in parallel have equivalent resistance equal to half one resistor.
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Kirchhoff’s first law is based on conservation of:
Reasoning: Current conservation comes from conservation of electric charge.
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The parallel resistance formula shows that adding resistors in parallel:
Reasoning: Additional branches provide easier paths for current flow.
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For three equal resistors $\mathbf{R}$ connected in parallel, the equivalent resistance is:
Reasoning: $\frac{\mathbf{1}}{\mathbf{R}_{\mathbf{eq}}}\mathbf{=}\frac{\mathbf{3}}{\mathbf{R}}$ $\mathbf{\Longrightarrow}$ $\mathbf{R}_{\mathbf{eq}}\mathbf{=}$ $\frac{\mathbf{R}}{\mathbf{3}}$
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In a parallel circuit, the voltage across each resistor is:
Reasoning: Each branch is directly connected across the source terminals.
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A resistor network has an equivalent resistance smaller than every individual resistor. This indicates:
Reasoning: Only parallel combinations produce an equivalent resistance lower than the smallest branch resistance.
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The combined resistance of parallel resistors represents:
Reasoning: Equivalent resistance replaces the parallel network while maintaining the same voltage-current relationship.
30 / 30
The formula for parallel resistance is obtained by comparing:
Reasoning: Kirchhoff’s first law gives total current, and Ohm’s law converts currents into resistance relationships.
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