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The circular loop PQRSTP formed by two c...

The circular loop PQRSTP formed by two circular segments of radii `R_1` and `R_2`, carries a current `I`. What is the magnetic field induction at the centre O? What will be the magnetic field induction at O if `alpha=90^@`?

Text Solution

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Let `hatn` be the unit vector perpendicular to the plane of loop downwards. Total magnetic field induction at O due to current through circular loop is
`vecB=vecB_(PQ)+vecB_(QR)+vecB_(RS)+vecB_(STP)`
`=0+(mu_0)/(4pi)I/R_2alphahatn+0+(mu_0)/(4pi)I/R_1(2pi-alpha)hatn`
`=(mu_0I)/(4pi)[alpha/R_2+((2pi-alpha))/(R_1)]hatn`
or `B=(mu_0I)/(4pi)[(alpha)/(R_2)+((2pi-alpha))/(R_1)]`
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If `alpha=90^@`, then
`B=(mu_0I)/(4pi)[(pi//2)/(R_2)+(3pi//2)/(R_1)]`
`=(mu_0I)/(8)[(1)/(R_2)+(3)/(R_1)]`
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