એક પાતળી ધાતુની પટ્ટી કાગળના સમતલને લંબ \(v\) વેગથી ગતિ કરે છે.આકૃતિમાં દર્શાવ્યા પ્રમાણે ચુંબકીયક્ષેત્ર \(B\) કાગળના સમતલની અંદરની દિશામાં પ્રવર્તે છે.જો પટ્ટીની ડાબી અને જમણી સપાટી પર પ્રેરિત થતી વિજભારઘનતા \({\sigma _1}\) અને \({\sigma _2}\) હોય તો..... (ફ્રિન્જ અસરને અવગણો)

A \({\sigma _1} = \frac{{ - { \in _0}\,vB}}{2}\,,\,{\sigma _2} = \frac{{{ \in _0}\,vB}}{2}\)
B \({\sigma _1} = { \in _0}\,vB\,,\,{\sigma _2} = - { \in _0}\,vB\)
C \({\sigma _1} = \frac{{{ \in _0}\,vB}}{2}\,,\,{\sigma _2} = \frac{{ - { \in _0}\,vB}}{2}\)
D \({\sigma _1} = {\sigma _2} = { \in _0}\,vB\)

Verified Solution

Answer & Solution

Correct Answer

(B) \({\sigma _1} = { \in _0}\,vB\,,\,{\sigma _2} = - { \in _0}\,vB\)

Step-by-step Solution

Detailed explanation

\(\because F=q E\) and \(F=q v B\) \(\therefore \mathrm{E}=\mathrm{vB}\) And Gauss's law in Electrostatics \(\mathrm{E}=\frac{\sigma}{\varepsilon_{0}}\) \(\mathrm{E}=\frac{\sigma}{\varepsilon_{0}}=\mathrm{vB} \Rightarrow \sigma=\varepsilon_{0} \mathrm{vB}\)…

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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