How Does Point Connection Affect Potential in a Circuit with Resistor Ratios?

[Peter Velkov]

Given an electric circuit, with resistors R₁ and R₂, find the equivalent resistance R.
Data: R₁ = √2.R₂

Points M and N have the same electric potential so they can be connected with a conductor and this way we get the second circuit.

 

[mfb]

Points M and N have the same electric potential so they can be connected with a conductor and this way we get the second circuit.

There is more you can do to simplify the circuit.

 

[Peter Velkov]

In terms of what? I know i can solve the parallel ones below, however that won't solve it. There should be a way to proof that M and N are also equal to O.

 

[mfb]

Sure, use symmetry again.

 

[Peter Velkov]

Sure, use symmetry again.

It's symmetrical along MO, NO but that doesn't help us much.

 

[gneill]

Temporarily remove the two vertical R2's. What are the potentials at M,N,O?

 

[Peter Velkov]

Temporarily remove the two vertical R2's. What are the potentials at M,N,O?

On the upper side we will have a resistor with √2R₂, but on the down side R₂/√2. So they will be different.

 

[gneill]

On the upper side we will have a resistor with √2R₂, but on the down side R₂/√2. So they will be different.

Really? What was your argument for M and N having the same potential in your first post?

 

[Peter Velkov]

Really? What was your argument for M and N having the same potential in your first post?

Equal amounts of charge will flow through the upper or downer R₁, and since the resistance is equal the voltage will be too. However I don't understand why the points O and M or N will have the same potential as the resistance along the path is different.

 

[gneill]

Equal amounts of charge will flow through the upper or downer R₁, and since the resistance is equal the voltage will be too. However I don't understand why the points O and M or N will have the same potential as the resistance along the path is different.

The potentials at M,N, and O are due to the voltage dividers comprised of the resistances in their branches. Here's the situation:

All three branches have the same potential difference across them (whatever you imagine to be placed across terminals AB). All three branches have the same resistance ratios...