(C) The magnetic field $B$ at the axial midpoint of a finite solenoid is given by the formula:$B = \frac{\mu_0 n i}{2} (\cos \theta_1 + \cos \theta_2)

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$\frac{N\mu_0 i}{\sqrt{4r^2 + l^2}}$

(C) The magnetic field $B$ at the axial midpoint of a finite solenoid is given by the formula:$B = \frac{\mu_0 n i}{2} (\cos \theta_1 + \cos \theta_2)$For the midpoint,$\theta_1 = \theta_2 = \theta$. Thus,$B = \mu_0 n i \cos \theta$.Here,$n = \frac{N}{l}$ is the number of turns per unit length.From the geometry of the solenoid,$\cos \theta = \frac{l/2}{\sqrt{r^2 + (l/2)^2}} = \frac{l}{\sqrt{4r^2 + l^2}}$.Substituting these values into the expression for $B$:$B = \mu_0 \left(\frac{N}{l}\right) i \left(\frac{l}{\sqrt{4r^2 + l^2}}\right)$$B = \frac{N \mu_0 i}{\sqrt{4r^2 + l^2}}$

Class 12 Physics Moving Charges and Magnetism Ampere’s circuital law and its application (Solenoid and Toroid)

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$A$ solenoid has $N$ turns,length $l$,and cross-sectional radius $r$. If a current $i$ flows through the solenoid,what is the magnetic field at the axial midpoint? (Given $l \simeq r$)

A
$\frac{N\mu_0 i}{\sqrt{l^2 + r^2}}$
B
$\frac{N\mu_0 i}{(4l^2 + r^2)^{3/2}}$
C
$\frac{N\mu_0 i}{\sqrt{4r^2 + l^2}}$
D
$\frac{N\mu_0 i}{(4r^2 + l^2)^{3/2}}$

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Moving Charges and Magnetism

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Ampere’s circuital law and its application (Solenoid and Toroid)

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$A$ solenoid has $N$ turns,length $l$,and cross-sectional radius $r$. If a current $i$ flows through the solenoid,what is the magnetic field at the axial midpoint? (Given $l \simeq r$)

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Verify Ampere's law for the magnetic field of a point dipole with dipole moment $\vec{M} = M\hat{k}$. Take $C$ as the closed curve running clockwise along the quarter circle of radius $R$ and center at the origin,in the first quadrant of the $x-z$ plane.

$A$ long solenoid of radius $1\,mm$ has $100$ turns per $mm$. If $1\,A$ current flows in the solenoid,the magnetic field strength at the centre of the solenoid is:

$A$ cylindrical conductor of radius $R$ is carrying a constant current. The plot of the magnitude of the magnetic field,$B$ with the distance $d$,from the centre of the conductor,is correctly represented by the figure:

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