Two long parallel wires in the yz plane at a distance 2a apart carry a steady current I in opposite directions. Midway between the wires is a rectangular loop of wire 2 b × d carrying current I as sho

Question

Two long parallel wires in the yz plane at a distance 2a apart carry a steady current I in opposite directions. Midway between the wires is a rectangular loop of wire $2 b \times d$ carrying current I as shown in the figure ( $b < a$ < 2b). The loop is free to rotate about the z axis and the currents remain fixed to I irrespective of the relative motion between the loop and the wire. The angle $θ$ when torque is max is $cos^{- 1} {\frac{xab}{b ^{2} + a ^{2}}}$ find x.

TestHub · Accepted Answer

cos ϕ = 2 ab b 2 + a 2 − r 2 ​ b 2 + a 2 − r 2 = 2 ab cos ϕ r 2 = b 2 + a 2 − 2 ab cos ϕ f = 2 π r k o ii d ​ a sin ( 90 + θ ) ​ = r sin ϕ ​ cos θ = r a sin ϕ ​ τ = F cos θ ⋅ b = F ⋅ r a sin ϕ ​ ⋅ b τ = 2 π r H 0 ​ i 2 d ​ ⋅ r a sin ϕ ​ ⋅ b = 2 π r 2 H 0 ​ i 2 a d b ​ sin ϕ τ = C ( b 2 + a 2 − 2 ab cos ϕ sin ϕ ​ ) d ϕ d τ ​ = ( b 2 − a 2 − 2 ab cos ϕ ) cos ϕ − sin ϕ ( − 2 ab sin ϕ ) 0 = ( b 2 + a 2 ) cos ϕ − 2 ab cos 2 ϕ − 2 ab sin 2 ϕ 0 = ( b 2 + a 2 ) cos ϕ − 2 ab ( cos 2 ϕ + sin 2 ϕ ) 0 = ( b 2 + a 2 ) cos ϕ − 2 ab cos ϕ = b 2 + a 2 2 ab ​

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