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Faraday's Law of induction stays that the induced emf is equal to negative rate of change of magnetic field connected with the circuit. Now let us consider a closed circuit made of wire with a resistor, a batter and a changing magnetic field that passes through the circuit. If there was no batter then by FLI

closed-line-integral(Electricfield)=-(rate of change of magnetic field)

Now if the batter is connected will the equation become

closed-line-integral(Electricfield)=-(rate of change of magnetic field)+(battery emf)?

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  • $\begingroup$ Your verbal description of the line integral form of FLI is incomplete. Please put it in mathematic terms using MathJax. $\endgroup$
    – Bob D
    Jul 19, 2021 at 16:19

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The formulae you are using are wrong

the emf produced in an LCR circuit is the emf produced in the inductor itself which opposes the change (Lenz Law) which is given by e=-A(dB/dt)

Now talking about the electric field part: Whenever there is a varying magnetic field in a cylindrical region, it forms circular loops of electric field and the direction of electric field at each point is tangential on the loop and as the electric field lines are forming closed loops, this electric field is a non conservative field

So basically both these formulae are used in different situations

You could've simply checked if your formula is correct or not by checking the dimensions Dimensions of electrical field and rate of change of magnetic flux are different

And note that emf in the inductor is rate of change of magnetic flux and not rate of change of magnetic field

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