The figure shows a cube made of wires each having a resistance R. The cube is connected into a circuit across a body diagonal AB as shown. Find the equivalent resistance of the network in this case.

Solution

Let us search for the points of same potential.

Since, the three edges of the cube from A viz., AC, AC₁ and AC₂ are identical in all respects, the circuit points C, C₁ and C₂ are at the same potential. Similarly, for the point B the sides BD, BD₁ and BD₂ are symmetrical and the points D, D₁ and D₂ are at the same potential.

As a next step, let us bring together the points C, C₁ and C₂ and bring together the point D, D₁ and D₂ as well. On redrawing the circuit into the plane we get,

Then, the cube will look as shown above in the figure.

Then,
The resistance of each edge is R.
For AC, 3 edges of the cube are connected in parallel.
Therefore,
Resistance between A and C =

\frac{R}{3}

Similarly,
Resistance between C and D =

\frac{R}{6}

and,

Resistance between D and B =

\frac{R}{3}

Now, AC, CD and DB are connected in series with each other.

Thus,
Equivalent resistance is equal to

\frac{R}{3} + \frac{R}{6} + \frac{R}{3}

R_equivalent=

\frac{5}{6} R .

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Current Electricity

The figure shows a cube made of wires each having a resistance R. The cube is connected into a circuit across a body diagonal AB as shown. Find the equivalent resistance of the network in this case.

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