Faraday's law in integral form is
(The symbols have their usual meaning)
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \( \int E . dl = \displaystyle\frac{-d\phi}{dt} \)
Faraday's law of induction says that the emf induced in a closed loop equals the negative rate of change of the magnetic flux through it. The emf is the line integral of the electric field around the loop.
\( \displaystyle \varepsilon = -\frac{d\phi_{B}}{dt} \)
\( \displaystyle \oint \vec{E}\cdot d\vec{l} = -\frac{d\phi_{B}}{dt} \)
Option (1) \( E . ds = \displaystyle\frac{Q}{\varepsilon_0} \) — Incorrect. This is Gauss's law for the electric field (flux of \( \vec{E} \) through a closed surface).
Option (2) \( \int E . dl = \displaystyle\frac{-d\phi}{dt} \) — Correct. The line integral of \( \vec{E} \) round a closed loop equals \( -d\phi/dt \): Faraday's law.
Option (3) \( \int B . ds = 0 \) — Incorrect. This is Gauss's law for magnetism: the net magnetic flux through a closed surface is zero.
Option (4) \( \int B . dl = \mu_0\left(i_c + \varepsilon_0 \displaystyle\frac{d\phi}{dt}\right) \) — Incorrect. This is the Ampere-Maxwell law, with the conduction current and the displacement current.