An infinite combination of resistors, each having resistance R=4 Ω , is given below. What is the net resistance between the points A and B ? (Each resistance is of equal value,R=4)
CONCEPT:
Resistance:
- The measurement of the opposition of the flow of electric current through a conductor is called the resistance of that conductor. It is denoted by R.
There are mainly two ways of the combination of resistances:
1. Resistances in series:
- When two or more resistances are connected one after another such that the same current flows through them are called as resistances in series.
- The net resistance/equivalent resistance (R) of resistances in series is given by:
- Equivalent resistance,
⇒ R = R1 + R2
2. Resistances in parallel:
- When the terminals of two or more resistances are connected at the same two points and the potential difference across them is equal is called resistances in parallel.
- The net resistance/equivalent resistance(R) of resistances in parallel is given by:
\(⇒ R = \frac{R_1R_2}{R_1+R_2}\)
Calculation:
Let R eq be the equivalent resistance of the network.
Hence above circuit can be summarised as
By using the above formula, eq. of R = 4Ω & Req
⇒ R eq = (4 × R eq ) / (4 + R eq )
Now, R & Req are in series. Therefore,
⇒ Net resistance = R + (4 × Req) / (4 + Req)
As, this is an infinite series. Hence, net resistance will be equal to Req
⇒ Req (4 + Req) = 4(4 + Req) + 4Req
⇒ Req2 + 4Req = 16 + 4Req + 4Req
⇒ Req2 + 4Req = 16 + 8Req
⇒ Req2 - 4Req - 16 = 0
Solving this quadratic equation:
⇒ Req = (-b ± √(b2 - 4ac)) / 2a
Where a = 1, b = -4, and c = -16:
⇒ Req = (4 ± √(16 + 64)) / 2
⇒ Req = (4 ± √80) / 2
⇒ Req = (4 ± 4√5) / 2
⇒ Req = 2 ± 2√5
Since resistance cannot be negative, we take the positive value:
∴ Req = 2 + 2√5
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