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(A) Consider a small element of the pipe subtending an angle $d	heta$ at the center. The current in this element is $dI = \frac{I}{2\pi} d	heta$.The

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$\frac{{\mu _0}I\sqrt 2 }{4{\pi ^2}R}$

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(A) Consider a small element of the pipe subtending an angle $d	heta$ at the center. The current in this element is $dI = \frac{I}{2\pi} d	heta$.The magnetic field $dB$ at the center due to this long current-carrying strip (acting as an infinite wire) is given by $dB = \frac{\mu_0 (dI)}{2\pi R} = \frac{\mu_0 I}{4\pi^2 R} d	heta$.The direction of $dB$ is perpendicular to the radius vector of the element. For the quarter portion,the angle $	heta$ ranges from $0$ to $\pi/2$. Resolving the field into components:$dB_x = dB \sin	heta$ and $dB_y = -dB \cos	heta$.Integrating these components from $0$ to $\pi/2$:$B_x = \int_0^{\pi/2} \frac{\mu_0 I}{4\pi^2 R} \sin	heta d	heta = \frac{\mu_0 I}{4\pi^2 R} [-\cos	heta]_0^{\pi/2} = \frac{\mu_0 I}{4\pi^2 R}$.$B_y = \int_0^{\pi/2} \frac{\mu_0 I}{4\pi^2 R} \cos	heta d	heta = \frac{\mu_0 I}{4\pi^2 R} [\sin	heta]_0^{\pi/2} = \frac{\mu_0 I}{4\pi^2 R}$.The magnitude of the resultant magnetic field is $B = \sqrt{B_x^2 + B_y^2} = \sqrt{(\frac{\mu_0 I}{4\pi^2 R})^2 + (\frac{\mu_0 I}{4\pi^2 R})^2} = \frac{\mu_0 I \sqrt{2}}{4\pi^2 R}$.

Column $I$	Column $II$	Column $III$
$(I)$ Electron with $\overrightarrow{v}=2 \frac{E_0}{B_0} \hat{x}$	$(i)$ $\overrightarrow{E}=E_0 \hat{z}$	$(P)$ $\overrightarrow{B}=-B_0 \hat{x}$
$(II)$ Electron with $\overrightarrow{v}=\frac{E_0}{B_0} \hat{y}$	$(ii)$ $\overrightarrow{E}=-E_0 \hat{y}$	$(Q)$ $\overrightarrow{B}=B_0 \hat{x}$
$(III)$ Proton with $\overrightarrow{v}=0$	$(iii)$ $\overrightarrow{E}=-E_0 \hat{x}$	$(R)$ $\overrightarrow{B}=B_0 \hat{y}$
$(IV)$ Proton with $\overrightarrow{v}=2 \frac{E_0}{B_0} \hat{x}$	$(iv)$ $\overrightarrow{E}=E_0 \hat{x}$	$(S)$ $\overrightarrow{B}=B_0 \hat{z}$

$A$ long thin-walled pipe of radius $R$ carries a current $I$ along its length. The current density is uniform over the circumference of the pipe. The magnetic field at the center of the pipe due to the quarter portion of the pipe shown is:

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