"Seven Deadly Sins:
Wealth without work
Pleasure without conscience
Science without humanity
Knowledge without character
Politics without principle
Commerce without morality
Worship without sacrifice."
– Mahatma Gandhi
The following multiple choice question included in the All India Engineering/Architecture Entrance Examination (AIEEE) 2011 will be interesting and useful to you:
A current I flows in an infinitely long wire with cross section in the form of a semi-circular ring of radius R. The magnitude of the magnetic induction along its axis is
(1) μ0 I/4πR
(2) μ0 I/π2R
(3) μ0 I/2π2R
(4) μ0 I/2πR
It would have been better if the magnitude of the magnetic induction at the axis was asked for since the words along the axis usually means directed along the axis. [The component of field directed along the axis is zero].
Forget about it. The question setter requires you to calculate the magnetic flux density at points on the axis (of the semicircular ring shaped cross section of the infinitely long wire). In the adjoining figure we have shown the cross section (of the given wire) lying in the XY plane. The length of the wire is along the Z-axis and the current in the wire is supposed to flow along the negative Z-direction. The broad infinitely long wire can be imagined to be made of a large number of infinitely long straight wire strips, each of small width dℓ.
With reference to the figure, we have dℓ = Rdθ.
The magnetic flux density due to the above strip is shown as dB1 in the figure. It has an X-component dB1 sinθ and a Y-component dB1cosθ. When we consider a similar strip of the same with dℓ located symmetrically with respect to the Y-axis, we obtain a contribution dB2 to the flux density. The flux density dB2 has the same magnitude as dB1. It has X-component dB2 sinθ and Y-component dB2cosθ. The X-components of dB1 and dB1 are of the same magnitude and direction and they add up. But the Y-components of dB1 and dB1 are in opposite directions and have the same magnitude. Therefore they get canceled. The entire conductor therefore produces a resultant magnetic field along the negative X-direction.
The wire strip of width dℓ cn be imagined to be an ordinary thin straight infinitely long wire carrying current Idℓ/πR since the total current I flows through the semicircular cross section of perimeter πR. Putting dB1 = dB2 = dB we have
dB = μ0 (Idℓ/πR)/2πR = μ0 (IRdθ/πR)/2πR = μ0Idθ/2π2R
The X-component of the above field is (μ0I/2π2R) sinθ dθ
The field due to the entire conductor is B = 0∫π [(μ0I/2π2R) sinθ]dθ
Or, B = μ0I/π2R since 0∫π sinθ dθ = 2
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