Class 10, Science, Chapter-11, Lecture-3, Combination of Resistances (Notes)
COMBINATION OF RESISTORS IN SERIES:
Two or more resistors that are connected end-to-end consecutively are called resistors connected in series.
LAWS OF COMBINATION OF RESISTORS IN SERIES:
- The resistance of the combination of resistors is equal to the sum of the individual resistors. ${\rm{R}} = {{\rm{R}}_1} + {{\rm{R}}_2} + {{\rm{R}}_3} + \cdots $
- In a circuit, if the resistors are connected in series, the current is the same in every part of the circuit.
- The total voltage across the combination is equal to the sum of the voltage drop across the separate resistors.
DERIVATION OF THE FORMULA ${\rm{R}} = {{\rm{R}}_1} + {{\rm{R}}_2} + {{\rm{R}}_3} + \cdots $
Let R1 , R2 , R3 be the three resistors in series,
and V1 , V2 , V3 be the potential differences across the resistors R1 , R2 , R3 respectively.
Let I be the current flowing in the wire,
V be the total potential difference across the combination , and
R be the equivalent resistance of the combination.
Therefore , V = I R
The current flowing through the circuit is I , therefore ,
V1 = I R1
V2 = I R2
V3 = I R3 and
V = I R
The total p.d. across the circuit is the sum of individual p.d. across each resistor , therefore ,
- $$V\,\,\, = \,\,\,{V_1}\,\,\, + \,\,\,{V_2}\,\,\, + \,\,\,{V_3}$$
- $$IR\,\,\, = \,\,\,I{R_1}\,\,\, + \,\,\,I{R_2}\,\,\, + \,\,\,I{R_3}$$
- $${R\,\, = \,\,\,{R_1}\,\,\, + \,\,\,{R_2}\,\,\, + \,\,\,{R_3}}$$
COMBINATION OF RESISTORS IN PARALLEL:
Two or more resistors that are connected between the same two points are called the resistors connected in parallel.
LAWS OF COMBINATION OF RESISTORS IN PARALLEL:
- The reciprocal of the equivalent resistance of the combination of resistors is equal to the sum of the reciprocals of separate resistances. ${1 \over {\rm{R}}} = {1 \over {{{\rm{R}}_1}}} + {1 \over {{{\rm{R}}_2}}} + {1 \over {{{\rm{R}}_3}}} + \cdots $
- In a circuit, if the resistors are connected in parallel, the current in various resistors are inversely proportional to the resistances. The total current is the sum of the currents flowing in the different resistors.
- The voltage across each resistor is equal to the voltage across the combination.
DERIVATION OF THE FORMULA ${1 \over {\rm{R}}} = {1 \over {{{\rm{R}}_1}}} + {1 \over {{{\rm{R}}_2}}} + {1 \over {{{\rm{R}}_3}}} + \cdots $
Let R1 , R2 , R3 be the three resistors in parallel,
and I1 , I2 , I3 be the current flowing in the resistors R1 , R2 , R3 respectively.
Let I be the total amount of current flowing in the circuit,
V be the potential difference across each resistance , and
R be the equivalent resistance of the combination.
Therefore, $$I\,\,\, = \,\,\,{V \over R}$$
The p.d. across each resistors is V , therefore ,
The total current flowing in the circuit is the sum of currents in each resistor , therefore ,
$${\rm{I}} = {{\rm{I}}_1} + {{\rm{I}}_2} + {{\rm{I}}_3}$$
$$ \Rightarrow {{\rm{V}} \over {\rm{R}}} = {{\rm{V}} \over {{{\rm{R}}_1}}} + {{\rm{V}} \over {{{\rm{R}}_2}}} + {{\rm{V}} \over {{{\rm{R}}_3}}} + \cdots $$
$$ \Rightarrow {{\rm{1}} \over {\rm{R}}} = {{\rm{1}} \over {{{\rm{R}}_1}}} + {{\rm{1}} \over {{{\rm{R}}_2}}} + {{\rm{1}} \over {{{\rm{R}}_3}}} + \cdots $$