Find heat produced on closing the switch S

[Sanchayan Dutta]

Homework Statement

Homework Equations

HEAT PRODUCED=WORK DONE BY BATTERIES - ENERGY ABSORBED BY CAPACITORS

The Attempt at a Solution

INITIALLY BEFORE SWITCH CLOSED :Initially a current flows only in the left most loop involving 4 microfarad capacitor.The charged on the capacitor during steady state is (20*4)=80 microcoulumbs.

FINALLY AFTER SWITCH IS CLOSED:Now a decaying current current flows in all the three loops -left,right, and bottom.However we are concerned only with steady state.So suppose the potential at the topmost junction (between the two capacitors and 2 ohm resistor) is x volts.Assuming that the node on the left (connected to 20 V cell,resistance R and resistance 4 ohm) is 0 volts.So we can write the nodal equation at x. (x-20)4+(x-0)5=0.Hence,x=80/9 volts.

Now work done by cell= (Charge Flown )* Cell Voltage = Final Charge On Left Plate Of Capacitor 4 microfarads - Initial Potential On Left Plate Of 4 microfarad= 4(20-80/9) - 4*20=-35.56 microjoule(negative).

Energy Absorbed By Capacitor = 1/2(4)(20-80/9)^2 + 1/2(5)(80/9)^2 - 1/2(4)(20)^2= -355.56 microjoule

HEAT PRODUCED=WORK DONE BY BATTERIES - ENERGY ABSORBED BY CAPACITORS=-35.56 + 355.56=320 microjoules

But this does not match with any of the options.Please help.

 

[cnh1995]

What is the energy stored in the circuit before the switch is closed? What will be the energy in the circuit (in steady state) after the switch is closed?

 

[Sanchayan Dutta]

Please read my solution which I posted.

 

[cnh1995]

Initially a current flows only in the left most loop involving 4 microfarad capacitor.The charged on the capacitor during steady state is (20*4)=80 microcoulumbs.

Right.

FINALLY AFTER SWITCH IS CLOSED:Now a decaying current current flows in all the three loops

Are you sure? Think in terms of the steady state.

 

[Sanchayan Dutta]

Is the final charge on 4 microfarad capacitor same as initial because no current flows finally in steady state?Does the other capacitor remain uncharged?

 

[cnh1995]

Is the final charge on 4 microfarad capacitor same as initial because no current flows finally in steady state?Does the other capacitor remain uncharged?

No current flows even in the transient state. You can verify this by applying KVL when the switch is closed.

 

[Sanchayan Dutta]

I agree that current will flow as transient current initially but finally the capacitors will behave as open circuits and no current will flow in any of the loops.There will be p.d. only across the left capacitor.

 

[cnh1995]

I agree that current will flow as transient current initially

After the switch is closed, there's no transient. And the transient before closing the switch should be ignored since the switch is open for a sufficiently long time.

 

[Sanchayan Dutta]

Ok I get that.So should the heat dissipated be 0 after switch is closed?

 

[cnh1995]

Ok I get that.So should the heat dissipated be 0 after switch is closed?

Yes. The 4uF capacitor retains all the charge and no current flows through the circuit after the switch is closed.

 

[Sanchayan Dutta]

Alright.Thank You.You are from india?Which university? =) Thanks a lot for your help!

 

[cnh1995]

Thank You

You're welcome!

Which university?

Pune..

 

[Sanchayan Dutta]

Nice to meet you.Bye :-)