Balanced length of a potentiometer wire

[kalupahana]

Homework Statement

A battery if emf E₀=12v connected across a 4m long uniform wire having resistance 4Ω. The cells of small emf E₁=2v and E₂=4v having internal resistance 2Ω and 6Ω respectively, are connected as shown in figure below. If galvanometer shows no deflection at the point N, then distance of N from point A is equal to

a. 1/6 cm
b. 25 cm
c. 75cm
d. 50 cm

Homework Equations

V=IR
Kirchoff's laws
V=kl (v=voltage, k=voltage per unit length, l=length) => for potentiometer

The Attempt at a Solution

Since the potentiometer is in balanced condition no current passes through galvanometer.
So applying V=IR for potentiometer
12v=(8+4)I
12=12I
I=1A
BY applying V=IR to potentiometer wire
V=4I=4.1
V=4v
Potential drop per unit length
k=4v/4m K=1 vm^-1

I have no idea to find potential of the points of circuit which is connected to the potentiometer. How to find that. When Kirchoff;s laws applied that potential get 0 and balanced length become 0.

Applying Kirchoff's Laws
4v=6I₁
I₁=2/3 A

-2v=2I₂
I₂=-1A
Direction of current is opposite

Now by V=IR, the potential of circuit become 0.
How to complete this.

 

[cnh1995]

I believe there is a mistake in the diagram. The middle wire( where i1-i2 is shown) should not be there. It connects one end of galvanometer to 12V directly. If it is there, the galvanomater will read 0 at point A.
If the middle wire is not there:
Assigning potentials to various points, let A be at 12V. So, B is at 8V and so on..
Applying KVL to the loop containing the two sources, it turns out that one end of the galvanometer(fixed end) is at 9.5V potential.
So, when the other (moving) end is also at 9.5V on the wire, galvanometer will read 0.
So, starting from point A(12V), N will be 2.5V away i.e. 2.5m away from point A (since potential gradient=1V/m).
So, the answer should be 250cm, which is none of the options.

 

[lychette]

I agree with CNH to some extent...it looks like a mistake in which case the voltage across the external circuit is 2.5V which equates to a section of wire 2.5m long (any section of wire 2.5m long would 'balance' 2.5V)
If the diagram is not wrong then there is a short circuit caused by that central connection and the voltage across the circuit is ZERO.
I think the 'best' answer is 1/6cm..1.3 mm...the wire is 4000mm long and 1.3mm is as close to zero as makes no difference.
If that is the answer then this is either a stupid question or a question designed to promote discussion.