Question

In a series combination of resistances
(A) Potential difference is same across each resistance
(B) Total resistance is reduced
(C) Current is same in each resistance
(D) All above are true.

Answer 1

Hint : Equivalent resistance of a series arrangement of resistor is the sum of the individual resistances in the circuit. On applying Kirchhoff's voltage rule around a loop with resistors in series, and investigating the result we will find the correct option.

Formula used: In this solution we will be using the following formula;
 $ {R_s} = {R_1} + {R_2} + ... + {R_n} $ where $ {R_s} $ is the equivalent resistance of a series arrangement of resistors, and $ {R_1}...{R_n} $ are the individual resistances of the resistors.
 $ \sum V = 0 $ where $ V $ are the individual voltage drop across the elements in a circuit.
 $ V = IR $ where $ I $ is the current through a resistor, and $ R $ is the resistance of the resistor.

Complete step by step answer
Let us assume that two resistors are placed in series to a voltage source. The voltage drop across each resistor is given by the Ohms law as
 $ V = IR $ where $ V $ is the voltage, $ I $ is the current through a resistor, and $ R $ is the resistance of the resistor.
Hence, applying the Kirchhoff’s voltage law around the circuit, we have
 $ V - {I_1}{R_1} - {I_2}{R_2} = 0 $
 $ \Rightarrow V = {I_1}{R_1} + {I_2}{R_2} $
The voltage $ V $ can be written as
 $ V = {I_s}{R_s} $
Hence, $ {I_s}{R_s} = {I_1}{R_1} + {I_2}{R_2} $
Now, we know that the equivalent resistance of a series arrangement of resistors is given as
 $ {R_s} = {R_1} + {R_2} + ... + {R_n} $ , where $ {R_1}...{R_n} $ are the individual resistances of the resistors. Hence, for the two resistors in series we assumed, we have
 $ {R_s} = {R_1} + {R_2} $
Hence, from $ {I_s}{R_s} = {I_1}{R_1} + {I_2}{R_2} $ we can get the equivalent resistance formula by making
 $ {I_s} = {I_1} = {I_2} $
Hence, the current flowing through a series combination of resistors is the same in each resistor.
Thus, the correct option is C.

Note
In actuality, the current through each resistance in a series combination is obtained from the Kirchhoff’s current law, which in turn is obtained from the principle of conservation of charge. From the current law, we know that the current flowing into a node is the same as the current flowing out the node. Hence, since in a series combination, there are no branches, then the current flowing out one resistor must be equal to the current flowing into the next one.