Q.A circular coil expands radially in a region of magnetic field and no electromotive force is produced in the coil. This can be because

As the coil's area grows, an emf appears unless the linked flux $\Phi=B_\perp A$ stays constant. If the field lies in the plane of the coil, its perpendicular component is zero, so $\Phi=0$ whether or not the field varies — no emf (b correct). If there is a perpendicular component but it decreases at just the right rate as $A$ increases, the product $B_\perp A$ stays constant — no emf (c correct …

Concept understanding. Faraday's law gives $\mathcal{E} = -d\Phi/dt$, with $\Phi = \int\vec{B}\cdot d\vec{A} = B_\perp A$, where $B_\perp$ is the component of $\vec{B}$ perpendicular to the plane of the coil. As the coil expands radially, $A$ increases; for zero emf the flux must not change with time.

Testing each option.

• (a) Field simply "constant": if it is constant and has a perpendicular component, then $\Phi = B_\perp A$ grows as $A$ increases $\Rightarrow$ emf is produced. Wrong.
• (b) Field lies in the same plane as the coil: its perpendicular component is zero, so $\Phi = 0$ at all times regardless of whether the field varies. $d\Phi/dt = 0 \Rightarrow$ no emf. Correct. …

Method: Diagnosing "No EMF Despite a Changing Area" — Check the PRODUCT, Not Either Factor Alone

Use this technique whenever a loop's area is changing (expanding, contracting, or otherwise deforming) but the question states that NO emf is produced — the trap is to check whether $B$ alone is constant, when the real condition is on the product $B_\perp A$.

Steps

Step 1: Write flux using only the perpendicular field component

where $B_\perp$ is the component of the field perpendicular to the loop's plane. A field component lying in the plane of the loop contributes exactly zero to the flux, no matter how large it is or whether it varies — this is easy to overlook.

Step 2: Recognise the general no-emf condition

With the loop's area $A$ changing, no emf requires

which can happen in (at least) two structurally different ways:

1. $B_\perp\equiv 0$ at all times — the field lies entirely in the loop's plane, so $\Phi=0$ identically, regardless of whether the field's in-plane magnitude varies or not;
2. $B_\perp\ne 0$ but decreases at exactly the rate needed to offset the growing (or shrinking) $A$, so the product stays fixed even though neither factor alone is constant. Step 3: Reject any option with a genuinely constant, nonzero $B_\perp$ …