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Question
Two long straight parallel current-carrying conductors are kept ‘a’ distant apart in the air. The direction of current in both the conductors is the same. Find the magnitude of force per unit length and the direction of the force between them. Hence define one ampere.
Solution
The magnetic induction B1 set up by the current I1 flowing in the first conductor at a point somewhere in the middle of the second conductor is
`B_1 = (mu_0I_1)/(2pia)` ...(1)
The magnetic force acting on the portion P2Q2 of length ℓ2 of the second conductor is
F2 = l2 ℓ2 B1 sin 90° ...(2)
From equation (1) and (2),
F2 = `(mu_0I_1I_2ℓ_2)/(2pia)`, towards first conductor
`F_2/ℓ_2 = (mu_0I_1I_2)/(2pia)` ...(3)
The magnetic induction B2 set up by the current I2 flowing in the second conductor at a point somewhere in the middle of the first conductor is
`B_2 = (mu_0I_2)/(2pia)` ...(4)
The magnetic force acting on the portion P1Q1 of length ℓ1 of the first conductor is
F1 = I1ℓ1B2 sin 90° ...(5)
From equation (3) and (5)
`F_1 = (mu_0I_1I_2ℓ_1)/(2pia)`, towards second conductor
`F_1/ℓ_1 = (mu_0I_1I_2)/(2pia)` ...(6)
The standard definition of 1A
If I1 = I2 = 1A
ℓ1 = ℓ2 = 1m
a = 1m in V/A then `F_1/ℓ_1 = F_2/ℓ_2 = (mu_0 xx 1 xx 1)/(2pi xx 1) = 2 xx 10^-7` N/m
∴ One ampere is that electric current which when flows in each one of the two infinitely long straight parallel conductors placed 1m apart in vacuum causes each one of them to experience a force of 2 × 10-7 N/m.
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