The magnetic field of a given length of wire carrying a current for a single turn circular coil at centre is B, then its value for two turns for the same wire when the same current passing through it is _________.

C

\(\begin{array}{l}\text { length }=\text { circumference } \\\therefore \quad l=2 \pi R=n \times 2 \pi r \\\therefore \quad R=n r\end{array}\)
Where,
\(R =\) radius of one turn circular loc
\(r=\) radius of \(n\) turn circular loop
Magnetic field at the centre of \(R\) radius loop
\(B_1=\frac{\mu_0 I}{2 R}=\frac{\mu_0 I}{2 n r} quad \quad \ldots \ldots(1)\\
(\because R =n r)\)
Magnetic field at the centre of a radius \(n\) turns loop
\(B_2=\frac{\mu_0 n I}{2 r} \quad \quad \ldots \ldots(2)\)
Now \(\frac{ B _2}{B_1}=\frac{\mu_0 n I }{2 r} \times \frac{2 n r}{\mu_0 I }\)
\(\begin{array}{ll}\therefore \frac{ B _2}{B_1}=n^2 \\ \therefore B_2=n^2 B_1 \\ \therefore B_2=(2)^2 B \\ \therefore B _2=4 B\end{array}\)