0.4 Magnetic field due to current in a circular wire  (Page 4/5)

Problem : A current 10 A flowing through a straight wire is split at point A in two semicircular wires of radius 0.1 m. The resistances of upper and lower semicircular wires are 10 Ω and 20 Ω respectively. The currents rejoin to flow in the straight wire again as shown in the figure. Determine the magnetic field at the center “O”.

Magnetic field due to current in wire

Solution : The straight wire sections on extension pass through the center. Hence, magnetic field due to straight wires is zero. Here, the incoming current at A is distributed in the inverse proportion of resistances. Let the subscripts “1” and “2’ denote upper and lower semicircular sections respectively. The two sections are equivalent to two resistances in parallel combination as shown in the figure. Here, potential difference between “A” and “B” is :

Currents in semicircular segments

$$V_{AB}=\frac{IX{R}_{1}X{R}_2}{\left(R_1+R_2\right)}=I_1{R}_1=I_2{R}_2$$ $$\Rightarrow I_1=\frac{IX{R}_2}{\left(R_1+R_2\right)}=\frac{10X20}{30}=\frac{20}{3}A$$ $$\Rightarrow I_2=\frac{IX{R}_1}{\left(R_1+R_2\right)}=\frac{10X10}{30}=\frac{10}{3}A$$

We see that current in the upper section is twice that in the lower section i.e. $I_1=2I_2$ . Also, the magnetic field is perpendicular to the plane of semicircular section (plane of drawing). The current in the upper semicircular wire is clockwise. Thus, the magnetic field due to upper section is into the plane of drawing. However, the current in the lower semicircular is anticlockwise. Thus, the magnetic field due to lower section is out of the plane of drawing. Putting θ = π for each semicircular section, the net magnetic field due to semicircular sections at “O” is:

$$B=\frac{{\mu }_0{I}_1\pi }{4\pi R}-\frac{{\mu }_0{I}_2\pi }{4\pi R}$$ $$\Rightarrow B=\frac{{\mu }_0{I}_1\pi }{4\pi R}-\frac{{\mu }_0{I}_1\pi }{8\pi R}=\frac{{\mu }_0{I}_1\pi }{8\pi R}$$ $$\Rightarrow B=\frac{{10}^{-7}X20}{3X8X0.1}=8.3X{10}^{-7}T$$

The net magnetic field is into the plane of drawing.