What Happens to the Current Through R1 When E2 is Short Circuited?

[subhradeep mahata]

Homework Statement

The above circuit shows two cells of emfs E₁ and E₂ and two resistors with resistances R₁ and R₂. If E₂ is short circuited, then what is the current through R₁?

Homework Equations

The Attempt at a Solution

First of all, i am not sure what it exactly means when a cell is short circuited, but i suppose it means no current flows through E2. Going by this notion, first we consider the case where nothing is short circuited and everything is normal. Then, i will apply KVL in three loops as:

In outer loop clockwise:
E₁-R₁i+E₂=0
or, E₁+E₂=R₁i .....(1)
Inner left loop clockwise:
E₁-R₁i-R₂i₁=0
or, E₁=R₁i+R₂i₁ .....(2)
Inner right loop clockwise:
E₂+R₂i₁=0
or, E₂=-i₁R₂ ......(3)

Now, from (3): i₁= -E₂/R₂ ...(4)
from (1): i=(E₁+E₂)/R₁ ...(5)
So, i-i₁= {R₂(E₁+E₂)}/R₁R₂ (Lets call this x)
The final circuit looks something like this:

Now, according to question, x=0.
So, on solving, we get E₁+E₂=-E₂R₁/R₂
Substituting this in (5), we get i= -E₂/R₂
That's my final answer, but my books says the correct answer is E₁/R₁
Also, my teacher said that since no current flows through E₂, the circuit has only E₁, R₁ and R₂. So, required current= E₁/R1+R2

I am terribly confused, Please help me out.

 

[ehild]

Short circuiting the battery means that its terminals are connected with a wire of zero resistance, as shown in the figure by the red line. There flows a big current through the battery, and the voltage is zero across the terminals.

 

[subhradeep mahata]

@ehild and @haruspex

So no current is flowing through R₂. Therefore, required current is E₁/R₁, right?

 

[haruspex]

@ehild and @haruspex
View attachment 236123
So no current is flowing through R₂. Therefore, required current is E₁/R₁, right?

Yes.

 

[subhradeep mahata]

Just one more query: when i say a resistor is short circuited, does that mean there is a zero resistor wire is connected across its ends as well?

 

[haruspex]

Just one more query: when i say a resistor is short circuited, does that mean there is a zero resistor wire is connected across its ends as well?

Yes.

 

[subhradeep mahata]

Okay, thanks.