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Question
A conductor has 3 segments; two straight and of length L each and a semicircular with radius R. It carries a current I. What is the magnetic field B at point P?
Options
`(μ_0I)/(4πR)`
`(μ_0I)/(4πR^2)`
`(μ_0I)/(4R)`
`(μ_0I)/(4π)`
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Solution
`(μ_0I)/(4R)`
Explanation:
Express the relation of force due to portion I and III.
F = ILB sin `theta` .....(i)
Substitute 180° for `theta` to find the force in the straight portion.
F = ILB sin(180°)
F = 0
Magnetic field in the portion I and III is zero because sin(180°) = 0.
Express the relation of the magnetic field for the circular wire.
DB = `mu_0/(4pi) (Idl)/(R^2)` .....(ii)
Integrate equation (ii).
`int dB = (mu_0 I)/(4pi R^2) int_0^(aR) dI`
B0 [B] = `(mu_0 I)/(4pi R^2)` 20 [I]
Simplify the above equation.
B = `(mu_0 I)/(2R)` ......(iii)
Portion II is a semicircle, hence divide equation (iii) by 1/2.
B = `1/2 ((mu_0 I)/(2R))`
B = `(mu_0 I)/(4R)`
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