The Young's modulus of brass and steel are `10 ^(11) Nm ^(-2) and 2 xx 10 ^(11) Nm ^(-2)` respectively. A brass wire and a steel wire of the same length extend by 1 mm, each under the same force. If radii of brass and steel wires are `R_(B) and R_(S)` respectively, then .. .

Options

A. `R _(S) = (R _(B))/(sqrt2)`
B. `R _(S) = (R _(B))/(2)`
C. `R _(S) = sqrt2 R _(B)`
D. `R _(S) = 4 R _(B)`

Answer

A

Explanation

Young.s modulus
`Y= ( F //A)/(Delta L //L)`
`therefore Y = (FL)/(A Delta L) = (FL )/(pi R ^(2) Delta L )`
`therefore Y prop (1)/(R ^(2)) [because F,L, Delta L and pi` are constant]
`therefore (Y_(S))/( Y_(B)) = ((R_(B))/(R _(S))) ^(2)`
`therefore (2 xx 10 ^(11))/( 10 ^(11)) = ((R _(B))/( R _(S))) ^(2)`
`therefore 2 = ((R _(B))/( R _(S)))^(2)`
`therefore sqrt2 = (R _(B))/( R _(S))`
`therefore R _(S) = (R _(B))/(sqrt2)`