How Accurately Can Induced EMF Represent Batteries in Circuit Problems?

[Jahnavi]

Homework Statement

Homework Equations

The Attempt at a Solution

Sorry for the unclear image .

EMF induced in the left loop = 8

Total Resistance of the left loop = 6 Ohms

Current will be induced in anticlockwise direction in the left loop of magnitude 4/3A.

EMF induced in the right loop = 4

Total Resistance of the left loop = 4 Ohms

Current will be induced in anticlockwise direction in the right loop of magnitude 1 A.

By superposing the two currents , current flowing in the middle branch is 1/3 Ohms from bottom to top .

But the answer given is different .

Am I making some mistake ?

 

[gneill]

Your solution misses out on the interaction of the induced potential differences on the BE branch. That wire segment is shared, so you need to somehow account for this. Just using the total resistance of each loop independently doesn't do it...

Your induced EMFs are good. So draw an equivalent circuit, inserting the EMFs as voltage sources in the loops. Then solve using your favorite method.

 

[cnh1995]

By superposing the two currents , current flowing in the middle branch is 1/3 Ohms from bottom to top .

Induced emf is tricky!
Check whether KVL holds for the two loops if I_BE=1/3 A.

 

[Jahnavi]

Your solution misses out on the interaction of the induced potential differences on the BE branch. That wire segment is shared, so you need to somehow account for this.

Should I break the induced EMF's in two parts in each loop ?

Is it that the 8V induced EMF acts as a battery of 4 V in the left loop ( except the middle branch ) and 4V in the middle branch ? In anticlockwise sense both batteries assist each other .

This means for the right loop , 4V induced emf will be divided as 8V battery in the right loop ( except middle branch ) and 4V in the middle branch (calculated earlier ). But these two will be of opposing polarity such that net induced EMF in right loop is still 4 V ?

 

[gneill]

Is it that the 8V induced EMF acts as a battery of 4 V in the left loop ( except the middle branch ) and 4V in the middle branch ?

Really not sure what you're getting at there. The left loop generates a an EMF of 8 V, the right loop one of 4 V. The middle branch sits between these loops so it gets a share of both loop's "inductions", so they will interact in that branch just like a shared branch in mesh analysis.

If you were to open the loops anywhere but in the EB shared branch (which you'd have to spend more time pondering the implication of), you'd find the induced potential difference for the given loop across the gaps. So just open the loops in convenient places and stick in your voltage sources. Be sure to get the polarities correct for driving the expected induced currents. For example:

 

[Charles Link]

@Jahnavi Suggestion is to write out the two voltage loop equations, and also a conservation of current equation at the branch. That will give you 3 equations and 3 unknowns. (The problem basically has 3 unknown currents). $\\$ Note: Writing a 3rd voltage loop equation around the perimeter would just be the sum of the first two equations. And yes, on this one, I did get the book's answer. $\\$ Editing: Additional note: Your voltage loop equations need to have the form $V=I_1 R_1+I_2 R_2 +I_3 R_3 +I_4 R_4$. (You may have e.g. $I_2=I_3$, etc. There are basically 3 separate currents in this problem). The currents you need to mark on the diagram with an arrow beforehand, and if the current is opposite the direction of the EMF in that loop, that $IR$ term picks up a minus sign.

 

[Jahnavi]

I do get the correct answer .

Thanks @gneill and @Charles Link

 

[Jahnavi]

So draw an equivalent circuit, inserting the EMFs as voltage sources in the loops.

gneill ,

Can induced EMF's be always represented as a battery in the circuit as in this problem ?Are there any exceptions ?

 

[gneill]

gneill ,

Can induced EMF's be always represented as a battery in the circuit as in this problem ?Are there any exceptions ?

I'm not aware of any exceptions for simple loop configurations as in this problem.