An air-cored solenoid with \(3 0 ~ c m\) length and area of cross section \(25 cm^2\) and number of turns 500 , carries a current of 2.5 A . The current is suddenly switched off in a brief time of \(10^{-3} s\). How much is the average back emf induced across the ends of the open switch in the circuit ? Ignore the variation in magnetic field near the ends of the solenoid?

A 6.54 V
B 65.4 V
C 654 V
D 0.654 V

Verified Solution

Answer & Solution

Correct Answer

(A) 6.54 V

Step-by-step Solution

Detailed explanation

A
Flux \(\phi_1=\) N B A
\(\phi_1= N \left(\frac{\mu_0 NI }{l}\right) A\)
Linkage Total Flux \(\phi=\frac{\mu_0 N^2 A}{l} l\)
When the current is turned off, the magnetic flux associated with the solenoid becomes zero.
\(\therefore \phi_2=0\)
Induced emf
\(|\varepsilon|=\frac{|\Delta \phi|}{\Delta t}\)
\(\begin{array}{l}=\frac{\left|\phi_2-\phi_1\right|}{\Delta t} \\ =\frac{\mu_0 N^2 AI }{l \Delta t} \\ 4 \pi \times 10^{-7} \times(500)^2\end{array}\)
\(\begin{array}{l}\therefore|\varepsilon|=\frac{\times 25 \times 10^{-4} \times 2.5}{30 \times 10^{-2} \times 10^{-3}} \\ \therefore|\varepsilon|=6.54 V\end{array}\)

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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