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Find the magnetic induction of the field at the point O of a loop with current I, whose shapeis ilustrated (a) in Fig. the radil a and bas welll as the anglevarphi are known, (b) In Fig , the radiusa and theside bare known. |
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Answer» Solution :(a) From the Biot-Savart law, `dB = (mu_(0))/(4pi) i ((d vec(i) xx vec(r)))/(r^(3))` So, magnetic field induction DUE to TEH segment 1 at `O`, `B_(1) = (mu_(0))/(4pi) (i)/(a) (2pi - varphi)` also `B_(2) = B_(4) = 0`, as d `vec(L) uarr uarr vec(r)` and `B_(3) = (mu_(0))/(4pi) (i)/(b) varphi` Hence, `B_(0) = B_(1) + B_(2) + B_(3) + B_(4)` `= (mu_(0))/(4pi) [(2pi - varphi)/(a) + (varphi)/(b)]` (b) Here, `B_(1) = (mu_(0))/(4pi) (i)/(a) (3pi)/(a). vec(B_(2)) = 0`, `B_(3) = (mu_(0))/(4pi) (i)/(a) sin 45^(@)`, `B_(4) = (mu_(0))/(4pi) (i)/(a) sin 45^(@)`, and `B_(5) = 0` So, `B_(0) = B_(1) + B_(2) + B_(3) + B_(4) + B_(5)` `(mu_(0))/(4pi) (i)/(a) (3pi)/(2) + 0 + (mu_(0))/(4pi) (i)/(a) sin 45^(@) + (mu_(0))/(4pi) (i)/(a) sin 45^(@) + 0` `= (mu_(0))/(4pi)i [(3x)/(2a) + (SQRT(2))/(b)]`  
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