readings in ammeter

Question

In the given circuit the ammeter A₁ and A₂ are ideal and ammeter A₃ has a resistance 1.9 x 10^-3 . Then show that reading from ammeter are 82/27 , 34/27 & 0 respectively . 

 

The circuit is on http://i.imgur.com/KeZszNM.jpg

Answer 1

Hi Samagra:

 

   Since A₁ and A₂ are ideal, it is clear that no current flows through the non-ideal A₃, as a path of (theoretical) zero resistance is preferred by all current to a path with any non-zero resistance.

 

With that in mind, we can define three currents, which I called i₁ (going through the 3 Ω resistor and 8 V battery), i₂ (going through the 15Ω resistor and 20 V battery), and i₄ (going through the 5Ω resistor and 10 V battery).  I chose i₄ instead of i₃ for the last to avoid any confusion about whether that current goes through A₃.

 

These are three unknowns, and I thus need three equations to solve for them.

 

With those definitions, and some guesses about which way the currents ran, I set up two cases of Kirchhoff's loop rule (one loop going from the center to the bottom left, up to the top, and back to center;  the other from center, to top, to bottom right, and back to center) and one case of Kirchhoff's junction rule (at the center).  With my guesses about current direction (I do not have to be correct in guessing current direction, I just need to be consistent in my directions when setting up the Kirchhoff equations), I obtained:

 

[1]  -3i₁ + 8 V + 5i₄ + 10 V = 0

[2]  -10 V - 5i₄ - 15i₂ + 20 V = 0

[3]  i₂ = i₁+i₄

 

Cross-solving using these three equations, I was able to get the listed answers for i₁ (the current read by A₁) and i₂ (the current read by A₂)

 

If you would like to go into more detail about how to set this up or work through it, I am available for online sessions.  Just let me know!