Long answer questions I — task. Science State Board, Class 10.

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1. With the help of a circuit diagram derive the formula for the resultant resistance of three resistances connected

a. in series and

b. in parallel.

a. Resistors connected in series:

In a series circuit, the electrical components are connected in a , one after the other. The electric charge in a series circuit can only flow in .

If the circuit is broken or disturbed at any point in the loop, the through the circuit, and hence electric appliances linked to it .

Let the resistances of three resistors be \(R_1\), \(R_2\) and \(R_3\), be connected in series, and 'I' be the current flowing through the circuit. According to Ohm's law, the potential differences \(V_1\), \(V_2\) and \(V_3\) across \(R_1\), \(R_2\) and \(R_3\), respectively, are given by

\begin{array}{l} V_{1} = i i_{i} \\ V_{2} = i i_{i} \\ V_{3} = i i_{i} \end{array}

The sum of the potential differences across the ends of each resistor is given by

V = i_{i} + i_{i} + i_{i}

Substituting the values of \(V_1\), \(V_2\) and \(V_3\) in the above equation, we get

V = i i_{i} + i i_{i} + i i_{i}

---- (eq. 1)

A single resistor that can effectively replace all the resistors in the electric circuit to maintain the same current is known as an .

Let \(R_S\) be the effective resistance of the series-combination of the resistors. Then, the (eq. 1) becomes

\(V\) \(=\) \(I\ R_S\)

On substituting the values of effective resistance \(V\) in (eq. 1), we get

R_{S} = i_{i} + i_{i} + i_{i}

In a series circuit, the effective or equivalent resistance is equal to the sum of the individual resistances of the resistors.

The equivalent resistance in a series combination is the highest of the individual resistances.

b. Parallel circuits:

Components in parallel circuits are connected to the source in . There are multiple paths for the electric charge to flow in a parallel circuit.

Even if the circuit is broken at any point in the loop, the current can flow through the circuit. Any electric appliances linked to the circuit . Hence, the electrical wiring in our houses is made of parallel circuits.

The sum of the individual currents in each parallel branch in a parallel circuit the main current flowing into or out of the parallel branches. The potential difference across separate parallel branches are .

Let the resistances of three resistors be \(R_1\), \(R_2\) and \(R_3\), be connected parallel across points \(A\) and \(B\). Let '\(I\)' be the current flowing through the circuit.

The current \(I\) starts from the positive terminal of the battery, reaches point \(A\). The current passes through the resistors \(R_1\), \(R_2\) and \(R_3\) divided into three branches \(I_1\), \(I_2\) and \(I_3\), respectively.

According to Ohm's law, the current \(I_1\), \(I_2\) and \(I_3\) are given as

\begin{array}{l} i_{1} = \frac{V}{i_{i}} \\ i_{2} = \frac{V}{i_{i}} \\ i_{3} = \frac{V}{i_{i}} \end{array}

Then, the total current passing through the circuit is

i = i_{i} + i_{i} + i_{i}

Substituting the values of \(I_1\), \(I_2\) and \(I_3\) in the above equation, we get

i = \frac{V}{i_{i}} + \frac{V}{i_{i}} + \frac{V}{i_{i}}

---- (eq. 1)

Let \(R_P\) be the effective resistance of the parallel combination of resistors in the circuit. Then, the (eq. 1) becomes

I = \frac{V}{R_{P}}

---- (eq. 2)

On combining (eq. 1) and (eq. 2), we get

\frac{i}{i_{i}} = \frac{i}{i_{i}} + \frac{i}{i_{i}} + \frac{i}{i_{i}}

Hence, the sum of the reciprocals of the individual resistances is resistance when a number of resistors are connected in parallel.

The effective or equivalent resistance is

\frac{R}{n}

when the '\(n\)' number of resistors having an equal resistances '\(R\)' are connected in parallel.

The above statement can be given in the form of an equation as

i_{i} = \frac{i}{i}

The equivalent resistance in a parallel combination is the lowest of the individual resistances.

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9. Long answer questions I

Exercise condition: