An infinite wire placed along z-axis, has current I 1 in positive z-direction. A conducting rod placed in xy plane parallel to y-axis has current I 2 in positive y-direction. The ends of the rod subtend angles of +30 o and -60 o at the origin with positive x-direction. The rod is at a distance 'a' from the origin. Find net force on the rod.

Step-by-step solution

x a = tan θ x = a tan θ 1 d x = a sec 2 θ d θ B = μ 0 4 π 2 I 1 r · · · 2 d F → = I 2 dx B sin θ - k ^ · · · 3 From (1),(2) & (3) d F → = I 2 asec 2 θ d θ . μ 0 4 π 2 I 1 a sec θ . sin θ = μ 0 4 π 2 I 1 I 2 sin θ sec θ d θ ( − k ^ ) F → = μ o 4 π 2 I 1 I 2 ∫ − 30 ∘ 60 ∘ tan θ d θ = μ o 4 π 2 I 1 I 2 [ − log e cos θ ] − 30 ∘ 60 ∘ ( − k ^ ) F → = μ o 4 π 2 I 1 I 2 [ − log e cos 60 ∘ + log e cos ( − 30 ∘ ) ] = μ o 4 π 2 I 1 I 2 [ log e 3 2 × 2 ] ( − k ^ ) F → = μ o 4 π I 1 I 2 l n 3 ( − k ^ )