Question

(a) Find the expression for the resistors connected in series and parallel.

(b) Find the expression for resistivity of a material.

Answer 1

(a) For series combination:

In a series connection, all resistors are connected end to end (lengthwise) with each other. In such connections, the total resistance of the circuit increases. The given figure shows three resistors of resistance R₁, R₂, and R₃ respectively connected in a series with each other and a battery of V volts has been connected. In series combination an equal amount of current flows in each component of the circuit. So applied voltage gets divided into small voltages depending on the value of resistances of the resistors, which are connected in series.

Suppose the potential drop across R₁ is V₁, R₂ is V₂ and R₃ is V₃ and if you add voltages V₁, V₂, and V₃ and compare it with V, you will find that the applied voltage V is equal to the algebraic sum of the divided voltages V₁, V₂, and V₃ respectively. Thus, V = V₁ + V₂ + V₃

For each resistor, we have Ohm’s law as:

V = IR_S

Or, V₁ + V₂ + V₃ = IR_S

Or, IR₁ + IR₂ + IR₃ = IR_S

Or, I (R₁ + R₂ + R₃) = IR_S

∴ R_S = R₁ + R₂ + R₃

Here, R_S is the equivalent resistance of all three resistors, when they are connected in a series.

For parallel combination:

In a parallel connection, all the resistors are connected sideways with each other. In such a connection, the total resistance of the circuit decreases. The given figure shows parallel combination of three resistors of resistances R₁, R₂, and R₃ connected between points M and N.

In parallel combination supplied voltage does not divide across resistors in the circuit while current driven through the battery divides into small magnitudes across each resistor.

Suppose the current across R₁ is I₁, R₂ is I₂ and R₃ is I₃ and if you add currents I₁, I₂, and I₃ and compare it with I, you will find that the total current (I) is equal to the sum of divided currents I₁, I₂, and I_3.. Thus,

I = I₁ + I₂ + I₃

Applying Ohm’s law,

V = IR_P (where R_P is equivalent resistance.)

Or, I =V / R_P

∴ I = I₁ + I₂ + I₃

Here, R_p is the equivalent resistance of all three resistors, when they are connected in a parallel.

(b) As we know that resistance R of a material depends on two factors - one is the length (l) and other one is area of cross section (A) of the material.