Question

Derive an expression for the effective resistance when three resistors are connected in (i) series (ii) parallel.

Answer 1

Series combination of resistors : If a number of resistors are joined end to end so that the same current flows through each of them in succession, then the resistors are said to be connected in series.
Series combination of resistors : If a number of resistors are joined end to end so that the same current flows through each of them in succession, then the resistors are said to be connected in series.
Laws of resistances in series:

(i) Current through each resistance is same
(ii) Total voltage across the combination = sum of the voltage drops
(iii) Voltage drop across any resistor is proportional to its resistance
(iv) Equivalent resistance = sum of the individual resistances
(v) Equivalent resistance is larger than the largest individual resistance.
Derive an expression for the combination of three resistors connected in parallel.

Parallel combination of resistors : If a number of resistors are connected in
between two common points so that each of them provides a separate path for current, then they are said to be connected in parallel.

As shown in the figure, consider three resistors R1, R2, R3 connected in parallel.
Suppose a current I flows through the circuit when a cell of voltage V is connected across the combination. The current I at point A is divided into three parts I1, I2, I3 through the resistors R1, R2, R3 respectively. These three parts recombine at point B to give same current I.

∴ I = I1 + I2 + I3

As all the three resistors have been connected between the same two points A and B, voltage V across each of them is same. By Ohm’s law,

If RP be equivalent resistance of parallel combination, then,

Laws of resistance in parallel:

(i) Voltage across each resistor is same and is equal to the applied voltage.
(ii) Total current = sum of the currents through the individual resistances.
(iii) Currents through various resistance are inversely proportional to the individual resistances.
(iv) Reciprocal of equivalent resistance = sum of reciprocals of individual resistances.
(v) Equivalent resistance is less than the smallest individual resistance.

Answer 2

Answer:

cWe are asked to derive the expression for effective resistance when three resistors are connected in series as well as parallel.

Explanation:

                       We know that V = IR

where, V = voltage, I = Current and R = Resistance.

Series combination:

In Series combination, we know that distribution of voltage occurs across each resistor. Let the voltage across each resistor R₁, R₂ and R₃ be V₁, V₂ and V₃.

Hence, total voltage would be;

            V = V₁+V₂+V₃

V₁ = IR₁, V₂ = IR₂ and V₃ = IR₃

Hence, IR = IR₁+IR₂+IR₃

            IR = I(R₁+R₂+R₃)

This gives,   R = R₁+R₂+R₃  for series combination.

Parallel combination:

In parallel combination, we know that distribution of current occurs across each resistor. Let Current across each resistor R₁, R₂ and R₃ be I₁, I₂ and I₃.

Hence, total current would be:

                 I = I₁+I₂+I₃

$\frac{V}{R}$ = $\frac{V}{R1}$ + $\frac{V}{R2}$ + $\frac{V}{R3}$

$\frac{V}{R}$ = V($\frac{1}{R1}$ + $\frac{1}{R2}$ + $\frac{1}{R3}$)

For parallel combination,  

              $\frac{1}{R}$ = $\frac{1}{R1}$ + $\frac{1}{R2}$ + $\frac{1}{R3}$