A long current wire is bent in the shapes shown in figure. The circular portion has radius `R`. The magnetic induction at the center of the circular segment is

Options

A. `-(mu_(0)I)/(4piR)[2hat(k)p+pihat(i)],-(mu_(0)I)/(4piR)[hat(k)+(pi+1)hat(i)],-(mu_(0)I)/(4piR)[hat(k)+hat(j)+(3pi)/(2)hat(i)]`
B. `(mu_(0)I)/(4piR)[2hat(k)p+pihat(i)],-(mu_(0)I)/(4piR)[hat(k)+(pi+1)hat(i)],(mu_(0)I)/(4piR)[hat(k)+hat(j)+(3pi)/(2)hat(i)]`
C. `-(mu_(0)I)/(4piR)[2hat(k)p+pihat(i)],(mu_(0)I)/(4piR)[hat(k)+(pi+1)hat(i)],(mu_(0)I)/(4piR)[hat(k)+hat(j)+(3pi)/(2)hat(i)]`
D. `(mu_(0)I)/(4piR)[2hat(k)p+pihat(i)],(mu_(0)I)/(4piR)[hat(k)+(pi+1)hat(i)],(mu_(0)I)/(4piR)[hat(k)+hat(j)+(3pi)/(2)hat(i)]`

Answer

a

Explanation

`(a)` Figure `-(d)` shows magnetic field lines of a long wire. Note the direction of magnetic induction at points`P,Q,R` and `S`. From right `-` hand thumb rule the magnetic fields at points `P,Q,R` and `S` lie along the unit vectors `(hat(j)),(-hat(k)),(-hat(j))` and `(hat(k))` respectively.
Magnetic field due to segment ,`AB,`
`overset(rarr)(B_(1))=(mu_(0)I)/(4piR)(-hat(k))`
Magnetic field due to circular segment `BCD`,
`overset(rarr)(B_(2))=(mu_(0)I)/(2R)xx((pi)/(2pi))(-hat(i))`
Magnetic field due to segment `DE`,
`overset(rarr)(B_(3))=(mu_(0)I)/(4piR)(-hat(k))`
Resultant magnetic field,
`vec(B_(3))=vec(B_(1))+vec(B_(2))+vec(B_(3))" "=-(mu_(0)I)/(4piR)[2hat(k)+pihat(j)]`
`(b)` Magnetic field due to segment `AB`,
`vec(B_(1))=(mu_(0)I)/(4piR)(-hat(k))`
Magnetic field due to circular segment BCD,
`vec(B_(2))=(mu_(0)I)/(2R)((pi)/(2pi))(-hat(i))`
Magnetic field due to segment DE,
`vec(B_(3))=(mu_(0)I)/(4pir)(-hat(k))`
Resultant magnetic field
`vec(B_(R))=vec(B_(1))+vec(B_(2))+vec(B_(3))=-(mu_(0)I)/(4piR)[hat(k)+(pi+1)hat(i)]`
`(c)` Magnetic field due to segment AB,
`vec(B_(1))=(mu_(0)I)/(4piR)(-hat(k))`
Magnetic field due to circular segment BCDE,
`vec(B_(2))=(mu_(0)I)/(2R)xx((3pi//2)/(2pi))(-hat(i))`
Magnetic field due to segment EF,
`vec(B_(3))=(mu_(0)I)/(4piR)xx(-hat(j))`
Resultant magnetic field.
`vec(B_(R))=vec(B_(1))+vec(B_(2))+vec(B_(3))=-(mu_(0)I)/(4piR)[hat(k)+hat(j)+(3pi)/(2)hat(i)]`