Two identical conducting wires A and B of same dimensions and sam… - Vidyadip

MCQ

Two identical conducting wires $A$ and $B$ of same dimensions and same material are bent in the form of circular coil. Wire $A$ consists of single turn whereas wire $B$ consists of $2\, turns$. Both these wires are then suspended in a uniform magnetic field with their planes parallel to the one another and same current is passed through them. Which statement is correct ?

✓
Couple on loop $A$ is more than on loop $B$
B
Couple on loop $B$ is more than on loop $A$
C
Couples on both the loop are equal
D
Couple on loop $B$ is twice that on loop $A$

✓

Answer

Correct option: A.

Couple on loop $A$ is more than on loop $B$

a
$\mathrm{C}=\mathrm{NiAB} \sin \theta$

$\mathrm{C} \propto \mathrm{N} \cdot \mathrm{A}$

Here $\quad \frac{\mathrm{N}_{\mathrm{A}}}{\mathrm{N}_{\mathrm{B}}}=\frac{1}{2}$

If $\mathrm{r}_{\mathrm{A}}$ and $\mathrm{r}_{\mathrm{B}}$ be the radii of the circular loops $\mathrm{A}$ and $\mathrm{B}$ respectively, then the length $\ell$ of both the wires,

${\ell=2 \pi \mathrm{r}_{\mathrm{A}}=2 \times 2 \pi \mathrm{r}_{\mathrm{B}}}$

${\mathrm{r}_{\mathrm{A}}=2 \mathrm{r}_{\mathrm{B}}}$

Their cross-sectional areas $\mathrm{A}_{\mathrm{A}}$ and $\mathrm{A}_{\mathrm{B}}$ are in the ratio

$\frac{\mathrm{A}_{\mathrm{A}}}{\mathrm{A}_{\mathrm{B}}}=\frac{\pi \mathrm{r}_{\mathrm{A}}^{2}}{\pi \mathrm{r}_{\mathrm{B}}^{2}}=4$

Now, the ratio of couples on the two coils is

${\frac{C_{A}}{C_{B}}=\frac{N_{A}}{N_{B}} \frac{A_{A}}{A_{B}}=\frac{1}{2} \times 4=2} $

${C_{A}=2 C_{B}}$

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