Question

A cell arid two resistors R₁ and R₂ are provided to you.

1. Draw different combinations of resistors using R₁ , R₂ and the cell.

2. Derive an expression for the effective resistance of the circuit in which current is the same in both resistors

3. lf R₁ = 4Ω and R₂ = 6Ω, in which combination effective resistance is minimum? Find its value?

Answer 1

1. 

2. Derive an expression for effective resistance in series:

Consider three resistors R₁ , R₂ and R₃ connected in series and a pd of V is applied across it.

In the circuit shown above the rate of flow of charge through each resistor will be same i.e. in series combination current through each resistor will be the same. However, the pd across each resistor are different and can be obtained using ohms law.

pd across the first resistor V₁ = I R₁

pd across the second resistor V₂ = I R₂

pd across the third resistor V₃ = I R₃

If V is the effective potential drop and R is the effective resistance then effective pd across the combination is

V = IR

Total pd across the combination = the sum pd across each resistor, V = V₁ + V₂ + V₃

Substituting the values of pds we get IR = IR₁ + IR₂ + IR₃

Eliminating I from all the terms on both sides we get

R = R₁ + R₂ + R₃ ………(1)

Thus the effective resistance of series combination of a number of resistors is equal to the sum of resistances of individual resistors.

3. The effective resistance becomes minimum in parallel connection.

\(\frac{1}{R}=\frac{1}{R_1}+\frac{1}{R_2}\)

\(\frac{1}{R}\) = \(\frac{1}{4}+\frac{1}{6}\)

R = 2.4Ω