A circular coil connected to a battery of emf E produced a certain magnetic induction field at its centre. The coil is unwound, stretched to double its length rewound into a coil of `(1^(rd))/(3)` of the original radius and connected to a battery of emf E to produce same field at the centre. Then , E is

Options

A. `(2E)/(9)`
B. `(3E)/(7)`
C. `(9E)/(4)`
D. `(E)/(6)`

Answer

A

Explanation

Magnetic field at centre of a circular coil of n turns,
`B = (mu_(0)i)/(2r) n`
If wire is stretched to double it.s length,
`rArr 2 (2 pi n) = 2pir. n.`
or `n. = (2r)/(r.) n = (2r)/((r//3))n` ( `:. r .= (r)/(3)` )
`n. = 6n`
for same field at the centre,
`rArr (mu_(0) i n)/( 2r) = (mu_(0) i.n.)/(2r.)`
`rArr (i n)/( r) = (.(6n))/((r //3)) rArr i. = (i)/(8)`
`:.` volume remains same after double coil.s length,
`rArr lxx A = 2l xx A.`
`rArr A. = (A)/(2)`
Resistance, `R = (rhol)/(A)`
after double coil.s length , R. = `(rho (2l))/((A//2)) = (4 rhol)/(A) = 4R`
Hence, emf E. = `i.R. = ((i)/(18)) (4R) = (2)/(9) E ` (`:. ` E = iR)