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Question: A current i flows in a circular arc of wire whose radius is R, which subtends an angle of \(\dfrac{3...

A current i flows in a circular arc of wire whose radius is R, which subtends an angle of 3π2\dfrac{3\pi}{2}radian at its centre. The magnitude of electric field at the centre is:
A.μ∘iR\text{A.}\quad \dfrac{\mu_\circ i}{R}
B.μ∘i2R\text{B.}\quad \dfrac{\mu_\circ i}{2R}
C.2μ∘iR\text{C.}\quad \dfrac{2\mu_\circ i}{R}
D.3μ∘i8R\text{D.}\quad \dfrac{3\mu_\circ i}{8R}

Explanation

Solution

Magnetic field is the space around the magnet where the effect of magnet can be felt by another magnet or iron piece. Magnetic fields can also be produced by a moving charge whose intensity can be determined by the velocity and magnitude of charge. The S.I unit of magnetic field is Tesla (T) whereas the C.G.S unit is Gauss (G). The intensity of the magnetic field could be determined by applying Biot-savart law.
Formula used:
B=μ∘i2RB = \dfrac{\mu_{\circ}i}{2R }

Complete answer:
Since moving charge produces magnetic field, hence current, which is also the flow of electrons, produces magnetic field around it. Due to a complete circular loop, the magnetic field at its centre is given byB=μ∘i2RB = \dfrac{\mu_{\circ}i}{2R } where μ∘\mu_\circ is the permeability of free space. ‘R’ is the radius of a circular loop in which current ‘i’ is flowing.
Now, since magnetic field is linearly related to current, hence we can use unit factor method to find magnetic field due to a part of circular loop which follows:
Magnetic field at the centre of loop due to wire of angle 2π2\pi; B=μ∘i2RB = \dfrac{\mu_{\circ}i}{2R }
Hence magnetic field at the centre of loop due to wire of unit angle; B=μ∘i4πRB = \dfrac{\mu_{\circ}i}{4\pi R }
Hence magnetic field at the centre of loop due to wire of angle 3π2=3π2×μ∘i4πR=3μ∘i8R\dfrac{3\pi}{2} = \dfrac{3\pi}{2} \times \dfrac{\mu_{\circ }i}{4\pi R} = \dfrac{3\mu_{\circ}i}{8 R}

So, the correct answer is “Option D”.

Note:
One can also find out this value of the magnetic field by using integration. The original formula of magnetic at the centre of the wire due to complete loop is also obtained by using integration and applying Biot savart’s law. The termμ∘\mu_{\circ}, which is the permeability of free space has value 4π×10−7m kgs−2A−2=4π×10−7H/m4\pi \times 10^{-7}m \ kgs^{-2}A^{-2} = 4\pi \times 10^{-7} H/m, where H stands for henry, the unit of inductance.