Question

find the equivalent resistance of a cube in a circuit from 1 to 2

Answer 1

Answer:

Explanation:

Given: Circuit diagram of resistor connect in a network like cube.

To find: Equivalent resistance of the network.

Solution:Traditional method to solve this type of problem is to convert this network in a simplified network.See in attachment

1. All the resistance connected with point 1 (marked a,b and c) are in parallel.
2. All the resistance connected with point 2(marked d,e and f) are also in parallel.
3. All the in between resistances (marked as g,h,i,j,k and l) are in parallel connection.
4. All 3 cases are connected in series.

Step 1: Apply the formula of parallel connection for case 1

$\frac{1}{r_x} = \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \\ \\ \frac{1}{r_x} = \frac{1}{R} + \frac{1}{R} + \frac{1}{R} \\ \\ r_x = \frac{R}{3} \:ohm$

Step 2:Apply the formula of parallel connection for case 2

$\frac{1}{r_y} = \frac{1}{d} + \frac{1}{e} + \frac{1}{f} \\ \\ \frac{1}{r_y} = \frac{1}{R} + \frac{1}{R} + \frac{1}{R} \\ \\ r_y = \frac{R}{3} \:ohm$

Step 3:Apply the formula of parallel connection for case 3

$\frac{1}{r_z} = \frac{1}{g} + \frac{1}{h} + \frac{1}{i} + \frac{1}{j} + \frac{1}{k} + \frac{1}{l} \\ \\ \frac{1}{r_z} = \frac{1}{R} + \frac{1}{R} + \frac{1}{R} + \frac{1}{R} + \frac{1}{R} +\frac{1}{R}\\ \\ r_z = \frac{R}{6} \:ohm$

Step 4: As discussed above all three arrangements of parallel resistors are in series.

Thus,

$R_{1-2} = r_x + r_y + r_z \\ \\ R_{1-2} = \frac{R}{3} + \frac{R}{3} + \frac{R}{6} \\ \\ R_{1-2} = \frac{2R+2R+R}{6} \\ \\ R_{1-2} = \frac{5R}{6}\: ohm$

Final Answer:Equivalent resistance of the network is 5R/6 ohm.

Hope it helps you.

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