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Three resistors $R_{1,} R_{2,} and R_{3,}$ are connected in parallel and the combination is connected to a battery, ammeter, voltmeter, and key.

Which of the following the expression used for find the equivalent resistance of the combination of the resistors.

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\frac{1}{R} = \frac{1}{R_{1}} + \frac{1}{R_{2}} + \frac{1}{R_{3}}

R = R_{1} + R_{2} + R_{3}

R = R_{1} R_{2}

R = {R_{1} - R}_{2}

answer is A.

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Detailed Solution

The expression for equivalent resistance of three resistors

R_{1}, R_{2}

and

R_{3}

connected in parallel is

1 / R_{1} + 1 / R_{2} + 1 / R_{3}

It is discovered that the total current

I

is the sum of the individual currents flowing through each branch of the combination. For example, if three resistors are connected in parallel via voltage source V, then the total current in the circuit (I) is-

= > I = I_{1} + I_{2} + I_{3}

……………………….. (i) [

I = total current in the circuit]

When we apply Ohm's law to each resistor, we get

I_{1} = V / R_{1}

[

I_{1} = current in first resistance; V = voltage; R_{1} = first resistance]

….(ii)

I_{2} = V / R_{2}

[

I_{2} = current in second resistance; V = voltage; R_{2} = second resistance]

..(iii)

I_{3} = \frac{V}{R_{3}}

[

I_{3} = current in third resistance; V = voltage; R_{3} = third resistance]

….(iv)
Put the values of equations (ii),(iii) and (iv) in equation (i) we get

\frac{V}{R_{p}} = \frac{V}{R_{1}} + \frac{V}{R_{2}} + \frac{V}{R_{3}}

\frac{V}{R_{p}} = V [\frac{1}{R_{1}} + \frac{1}{R_{2}} + \frac{1}{R_{3}}]

\frac{1}{R_{p}} = \frac{1}{R_{1}} + \frac{1}{R_{2}} + \frac{1}{R_{3}}

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