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# Two unequal resistances R_{1} and R_{2} are connected across two identical batteries of emf ϵ and internal resistance r. Can the thermal energies developed in R_{1} and R_{2} be equal in a given time. If yes, what will be the condition?

Answer :

Given ∈ as emfs of the circuits with internal resistances r

Let the currents passing through the

resistance *R*_{1} and *R*_{2} be *i*_{1} and *i*_{2} , respectively for time t.

According to Kirchhoff’s Voltage Law, sum of

all voltages around any closed loop in a

circuit must equal zero.

ie,

Looking into the circuit 1-we get-

(1)

Similarly, the current in the other circuit,

(2)

The heat lost through a resistor r is given by

(3)

Now the total thermal energies through the resistances from Joule’s heating effect are given

by -

=

From(1), (2) and (3)

⇒( = (

⇒( = (

⇒( = (

Hence, the condition when the thermal energies developed in

R_{1}and R_{2} be equal in a given time is when the internal resistance r–

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