Parallel Circuits

Parallel connection → components placed next to each other in parallel + ends are connected like ladder.

• known as parallel circuit

Current

Important

In parallel circuit, total current flowing through into or out of the parallel branches is equal to the sum of the individual currents in each parallel branch $I=I_1+I_2$

Potential Difference

• across each branch must be same as electromotive force of the cell
• this is because cells are directly connected to the components.

Important

In parallel circuit, the p.d. across the parallel branches is the same. $V_{\mathcal{E}}=V_1=V_2$

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Since $I=I_1+I_2$: $I=\frac{V_1}{R_1}+\frac{V_2}{R_2}$ $I=V_{\mathcal{E}}(\frac{1}{R_1}+\frac{1}{R_2})$ $\frac{I}{V_{\mathcal{E}}}$ = $\frac{1}{R_1}+\frac{1}{R_2}$

We can define effective resistance $R_e$ such that $\frac{I}{V_{\mathcal{E}}}$ = $\frac{1}{R_e}$ = $\frac{1}{R_1}+\frac{1}{R_2}$

Reciprocal of effective resistance is the sum of the reciprocal of the individual resistances. simplifying the equation further: $\frac{1}{R_e}=\frac{R_1+R_2}{R_1{R}_2}$ $R_e=\frac{R_1{R}_2}{R_1+R_2}$

in special case where $R_1=R_2=R$ → applying above formula will give $R_2=\frac{R}{2}$ Addition → current in each branch of the parallel circuit is $\frac{I}{2}$

• this is intuitive → since we can expect that half current will flow in one branch and other half in other branch

In general if there are n number of parallel connections → then main current I is the sum of the individual currents flowing in the diff. branches. $I=I_1+I_2+....I_n$

voltage across these n number of branches are the same, that is $V_{\mathcal{E}}=V_1=V_2=V_n$

By combining this information: $I=\frac{V_1}{R_1}+\frac{V_2}{R_2}+\dots .+\frac{V_n}{R_n}$ $I=V_{\mathcal{E}}(\frac{1}{R_1}+\frac{1}{R_2}+\dots +\frac{1}{R_n})$ $\frac{I}{V_{\mathcal{E}}}=\frac{1}{R_1}+\frac{1}{R_2}+\dots +\frac{1}{R_n}$

Formulae

The effective resistance $R_e$ such that $R_e=\frac{V_{\mathcal{E}}}{I}$ becomes: $\frac{1}{R_e}=\frac{1}{R_1}+\frac{1}{R_2}+\dots +\frac{1}{R_n}$

REVISE, add your tag

In other words → effective resistance of all the resistors connected in parallel can be represented by. single resistor $R_e$ such that $R_e$ can be found using equation above

Important

In parallel circuit, the reciprocal of the effective resistance of resistors $\frac{1}{R_e}$ is equal to the sum of the reciprocal of the individual resistances.

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