JEE Mains · Physics · STD 12 - 3. current electricity

The resistance of an electrical toaster has a temperature dependence given by \(R\left( T \right) = {R_0}\left[ {1 + \alpha \left( {T - {T_0}} \right)} \right]\) in its range of operation. At \({T_0} = 300\,K,R = 100\,\Omega \) and at \(T = 500\,K,\,R = 120\,\Omega \). The toaster is connected to a voltage source at \(200\, V\) and its temperature is raised at a constant rate from \(300\) to \(500\, K\) in \(30\, s\). The total work done in raising the temperature is

A \(400\,\ln \,\frac{1.5}{1.3}\,J\)
B \(200\,\ln \,\frac{2}{3}\,J\)
C \(300\,J\)
D \(400\,\ln \,\frac{5}{6}\,J\)

Verified Solution

Answer & Solution

Correct Answer

(D) \(400\,\ln \,\frac{5}{6}\,J\)

Step-by-step Solution

Detailed explanation

\(R(T)=R_{o}\left(1+\alpha\left(T-T_{o}\right)\right)\) Applying boundary conditions, \(120=100(1+200 \alpha)\) \(\alpha=10^{-3}\, K^{-1}\) It is given that temperature increases at a constant rate from \(300\, \mathrm{K}\) to \(500\, \mathrm{K}\) in \(30 \,\mathrm{s}\). Hence,…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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