Question

please explain the above formula in detail.
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Answer 1

Hi my dear friend here is ur answer

When a bar magnet is bend into the shape of a horseshoe, the magnetic field between the poles is nearly uniform and usually quite strong. B = μ0I/(2πr). This equation is derived from one of Maxwell's equations called Ampere's law. Your fingers curl into the direction of the magnetic field produced by the current.

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Answer 2

Question:

To derive the formula of magnetic field due to a long straight current carrying conductor.

Answer:

Explanation:

↬ To derive this formula first we have to know Ampere's circuital law.

↬ Ampere's law state that the line integral of the magnetic field ($\vec{B}$) around any closed circuit is equal to μ₀ times the total current (I) passing through the closed circuit.

 where μ₀ is the magnetic constant.

  $\oint \vec {B}. \vec{dl} = \mu_0\:I$

↬ Ampere's circuital law is valid for any assembly of current.

Magnetic field due to a long straight current carrying wire:

↬ Consider a current carrying conductor XY.

↬ Consider a point P which is at a distance of r from the centre of the wire.

↬ Here we have to find the magnetic field at point P.

↬ Consider a small current element at P. Here the direction of $\vec{B}$ and dl are in the same direction. Hence the angle between them is zero.

↬ Hence the line integral of the closed loop is given by,

    $\oint \vec {B}. \vec{dl} = \oint B \:dl\:cos\theta$

↬ Here θ = 0

↬ Hence,

   $\implies \oint B\:dl$

   $\implies B \int dl$

↬ Here  $\int dl$ is the total length or the perimeter of the circular path.

↬ That is,

    $\int dl= 2 \pi \:r$

↬ Therefore

    $\oint \vec {B}. \vec{dl} = B\times 2\pi \:r$

↬ Now by Ampere's circuital law we know that,

   $\oint \vec {B}. \vec{dl} = \mu_0\:I$

↬ Hence,

   $B\times 2\pi r = \mu_0 I$

↬ Taking 2π r to the RHS side we get,

    $\boxed{B=\dfrac{\mu_0\:I}{2\pi\:r}}$

   where I = current through the conductor

              r = perpendicular distance from the wire

↬ The direction of magnetic field is given by the right hand thumb rule.