A long wire carrying a current i is bent to form a plane angle - Find the magnetic field B at point on the bisector of this angle situated at a distance x from the vertex-

Given that, #10; #10;Current =i #10; #10;Angle =α #10; #10;Magnetic field =B #10; #10;Now, magnetic field at a #10;point ...

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Byju's Answer

Standard X

Physics

Faraday's Experiment

A long wire c...

Question

A long wire carrying a current $i$ is bent to form a plane angle $α$ . Find the magnetic field $B$ at point on the bisector of this angle situated at a distance $x$ from the vertex.

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Solution

Given that,

Current $= i$

Angle $= α$

Magnetic field $= B$

Now, magnetic field at a point a from current carrying conductor making angle $θ_{1}$ and $θ_{2}$ with the ends of the wire is

$B = \frac{μ_{0} I}{4 π a} (sin θ_{1} + sin θ_{2})$

Now, $a = x sin \frac{θ}{2}$

$θ_{1} = \frac{θ}{2}$

$θ_{2} = 90^{0}$

Now, the magnetic field

$B_{l o w e r} = \frac{μ_{0} I}{4 π} ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \frac{1 + sin (\frac{θ}{2})}{x sin (\frac{θ}{2})} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦$

And $B_{u p p e r}$ is also same

No, if the current was flowing anticlockwise both these field would point upward outside the page

Now, net magnetic field

$B = B_{l o w e r} + B_{u p p e r}$

$B = 2 \times \frac{μ_{0} I}{4 π} ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \frac{1 + sin (\frac{θ}{2})}{x sin (\frac{θ}{2})} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦$

$B = \frac{μ_{0} I}{2 π x} ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \frac{1 + sin (\frac{θ}{2})}{sin \frac{θ}{2}} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦$

$B = \frac{μ_{0} I}{2 π x} ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \frac{{s i n (\frac{θ}{4}) + c o s (\frac{θ}{4})}^{2}}{2 sin (\frac{θ}{4}) cos (\frac{θ}{4})} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦$

Now, dividing numerator and denominator by ${cos}^{2} \frac{θ}{4}$

$B = \frac{μ_{0} I}{4 π x} ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \frac{{(1 + tan (\frac{θ}{4}))}^{2}}{tan \frac{θ}{4}} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦$

$B = \frac{μ_{0} I}{4 π x} [{(1 + tan (\frac{θ}{4}))}^{2}] cot (\frac{θ}{4})$

Now, the magnetic field is

$B = K cot (\frac{θ}{4})$

Where, $K = \frac{μ_{0} I}{4 π x} [{1 + tan (\frac{θ}{4})}^{2}]$

Hence, the magnetic field is $B = K cot (\frac{θ}{4})$

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A long wire carrying a current i is bent to form a plane angle α. Find the magnetic field B at point on the bisector of this angle situated at a distance x from the vertex.

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