Calculate magnetic induction at point O if the wire carrying a current I has the shape shown in (i) Fig.
Text Solution
Verified by Experts(i) Current carrying wire shown in figure can be considered in four parts; (ii) Circuit segment shown in figure can be considered in three parts
Sol . (i) Current carrying wire shown in figure can be considered in four parts.
(1) A straight part along y-axis.
Since, point O lies on its axis, therefore magnetic induction due to it is 
= 0.
(2) A semi-circular part in y-z plane.
Magnetic induction due to it is
= 
(– 
) = – 
(3) One fourth circle in x – y plane.
Magnetic induction due to it is
= 
(– 
)
= – 
(4) A straight part in x – y plane carrying current along negative y-directions.
Magnetic induction due to it is 
= 
(– 
) = – 
∴ 
= 
+ 
+ 
+ 
= – 
– 
( π + 2) 
(ii) Circuit segment shown in figure can be considered in three parts.
(1) A circular loop in y – z plane. Since, this loop is made of uniform wire, therefore, magnetic induction at O due to it is B 1 = 0.
(2) A straight part, parallel to x-axis.
Magnetic induction due to it is
B 2 = 
(– 
) = – 
.
(3) A straight part in y – z plane. Perpendicular distance of O from axis of this straight part is r = R cos 45º as shown in Fig. .
Fig.
Angles subtended by lines joining O and ends of this straight part with perpendicular drawn from O are α = – 45º and β = 90º.
Magnetic induction at O due to this part is B 3 = 
(sin α + sin β )
or 
= 
( 
– 1) 
∴ 
= 
+ 
+ 
= 
( 
– 1) 
– 
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