A wire loop formed by joining two semicircular sections of radii `R_(1) and R_(2)` and centre C carries a current I as shown in the figure . The resultant magnetic field at C has a magnitude of

Options

A. `(mu_(0)I)/(4)((1)/(R_(1))-(1)/(R_(2)))`
B. `(mu_(0)I)/(2)((1)/(R_(1))-(1)/(R_(2)))`
C. `(mu_(0)I)/(4)((1)/(R_(1))+(1)/(R_(2)))`
D. `(mu_(0)I)/(2)((1)/(R_(1))+(1)/(R_(2)))`

Answer

A

Explanation

Given :
Radius of inner semicircular loop ` = R _(1)` , Radius of outer semicircular loop `= R_(2)` and current in the loop = I . We know that magnetic field at the centre of current-carrying inner semi circular
Section `(B_(1)) = (1)/(2) xx (mu_(0)I)/(2R_(1)) = (mu_(0)I)/(4R_(1))`
Similarly, magnetic field at the centre of current-carrying outer semicircular section
`(B_(2)) = (mu_(0)I)/(4R_(2))`
Therefore resultant magnetic
field at `C = B_(1) - B_(2)`
` = (mu_(0)I)/(4R_(1))-(mu_(0)I)/(4R_(2))=(mu_(0)I)/(4)((1)/(R_(1))-(1)/(R_(2)))`