Moving Charges and Magnetism - Result Question 47
48. A coil of one turn is made of a wire of certain length and then from the same length a coil of two turns is made. If the same current is passed in both the cases, then the ratio of the magnetic inductions at their centres will be
(a) $2: 1$
(b) $1: 4$
(c) $4: 1$
(d) $1: 2$
[1998]
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Answer:
Correct Answer: 48. (b)
Solution:
- (b) Let $\ell$ be length of wire.
Ist case : $\ell=2 \pi r \Rightarrow r=\frac{\ell}{2 \pi}$
$B=\frac{\mu_0 I n}{2 r}=\frac{\mu_0 I \times 2 \pi}{2 \ell}=\frac{\mu_0 \pi I}{\ell}[\because n=1] \ldots(1)$
2nd Case $: \ell=2(2 \pi r^{\prime}) \Rightarrow r^{\prime}=\frac{\ell}{4 \pi}$
$B^{\prime}=\frac{\mu_0 I n}{2 \frac{\ell}{4 \pi}}=\frac{2 \mu_0 I \pi}{\frac{\ell}{2}}=4(\frac{\mu_0 \pi I}{\ell})=4 B$,
using (1) (where $n=2$ )
If a current carrying circular loop $(n=1)$ is turned into a coil having $n$ identical turns then magnetic field at the centre of the coil becomes $n^{2}$ times the previous field i.e., $B _{(n \text{ turn) }}=n^{2} B _{\text{(single turn) }}$
