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Physics Topics can also be used to explain the behavior of complex systems, such as the stock market or the dynamics of traffic flow.
What do you Mean by Equivalent Resistance of a Combination of Resistances?
Several resistances or resistors such as light bulbs, fans, pumps, etc. may be joined in a network variously for various purposes. Combination of resistances can be of three types:
- series combination,
- parallel combination and
- mixed combination.
Equivalent resistance: If a single resistance is used instead of a combination of resistances, to keep the current unchanged in the circuit, then that single resistance is called the equivalent resistance of the combination.
Series Combination of Resistances
A number of resistances are said to be connected in series if they are connected end to end consecutively so that the same current flows through each resistance when a potential differ-ence is applied across the combination.
Calculation of equivalent resistance: Three resistances R1, R2, R3 are connected in series in between the two points A and D of an electrical circuit [Fig.].
Let VA, VB, VC, and VD be the potentials at the points A, B, C and D respectively.
If I be the current flowing in the circuit, then according to Ohm’s law,
VA – VB = IR1 ……. (1)
VB – VC = IR2 ……. (2)
VC – VD = IR3 ……. (3)
Adding (1), (2) and (3) we have,
VA – VD = I(R1 + R2 + R3) …. (4)
If R be the equivalent resistance of the combination and if it is connected between the points A and D, the main current flowing in the circuit will remain the same. So
VA – VD = IR …… (5)
From (4) and (5) we have,
R = R1 + R2 + R3
Similarly, if n number of resistances are connected in series instead of the three resistances then,
R = R1 + R2 + R3 + ……. + Rn = \(\sum_{i=1}^n R_i\) ……..(6)
So, equivalent resistance of a series combination = sum of the individual resistances.
A few characteristics of series combination of resistances:
- The same current flows through each resistance.
- Total potential difference across the combination is equal to the sum of the individual potential difference across each resistance.
- Since current is constant, individual potential difference is directly proportional to the individual resistance.
Numerical Examples
Example 1.
Three resistances of magnitudes 20 Ω, 30 Ω and 40 Ω are connected in series,
(i) What is the equivalent resistance?
(ii) If the potential difference across the resistance 20 Ω is 1V, calculate the potential differences across the other two resistances and also the total potential difference across the combination.
Solution:
i) Equivalent resistance, R = 20 + 30 + 40 = 90 Ω
ii) For the resistance of 20 Ω,
current, I \(=\frac{\text { potential difference }}{\text { resistance }}\) = \(\frac{1}{20} \mathrm{~A}\)
Since it is a series combination, the current everywhere is \(\frac{1}{20} \mathrm{~A}\).
∴ Potential difference across 30 Ω resistance
= \(\frac{1}{20}\) × 30 = 1.5V
and across the 40 Ω resistance \(\frac{1}{20}\) × 40 = 2.0 V
Potential difference across the combination
= \(\frac{1}{20}\) × 90 = 4.5V
Example 2.
ρ1 and ρ2 are the reslstlvltles of the materials of two wires of the same dimensions. What will be the equivalent resistivity of the serles combination of the two wires? [Karnataka CET ‘03]
Solution:
Let l be the length and A be the cross sectional area of each wire.
The equivalent resistance in senes combination, is
R = R1 + R2 = \(\rho_1 \frac{l}{A}\) + \(\rho_2 \frac{l}{A}\) = (ρ1 + ρ2)\(\frac{l}{A}\) ……. (1)
In the series combination, length of the conductor = 2l; cross sectional area = A. Let the equivalent resistivity be ρ.
∴ R = \(\rho \frac{2 l}{A}\) ….. (2)
Comparing (1) and (2) we have,
\(\rho \frac{2 l}{A}\) = (ρ1 + ρ2)\(\frac{l}{A}\) or, ρ = \(\frac{\rho_1+\rho_2}{2}\)