# A flat circular coil of $n$ turns, area $A$ and resistance $R$ is placed in a uniform magnetic field $B$. The plane of coil is initially perpendicular to $B$. When the coil is rotated through an angle of $180_{o}$ about one of its diameter, a charge $Q_{1}$ flows through the coil. When the same coil after being brought to its initial position, is rotated through an angle of $360_{o}$ about the same axis a charge $Q_{2}$ flows through it. Then $Q_{2}/Q_{1}$

**A**

## $1$

**B**

## $2$

**C**

## $1/2$

**D**

## $0$

Hard

Solution

Verified by GMS

Correct option is D)

## Net charge flowing through the coil is given by $Q=RΔϕ $ where $R$ is the resistance

Initially the plane of ring is perpendicular to $B$, i.e Area vector $A$ is parallel to $B$ where $A$ is the area of the coil.

$∴$ initial flux $ϕ_{i}=A.B=AB$

Case 1) : Coil is rotated by $180_{o}$ i.e $A$ $∥$ $−B$

$∴$ Final flux $ϕ_{f}=−AB$

$⟹∣Δϕ∣=∣ϕ_{f}−ϕ_{i}∣=2AB$

Thus $Q_{1}=R2AB $ .............(1)

Case 2): Coil is rotated by $360_{o}$, i.e $A$ $∥$ $B$

Thus final flux $ϕ_{f}=AB$

$∴Δϕ=ϕ_{f}−ϕ_{i}=0$

Hence $Q_{2}=RΔϕ =0$

$⟹Q_{1}Q_{2} =0$

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