Question

A closely wound, circular coil with radius 2.80 cm has 720 turns. Part APart complete What must the current in the coil be if the magnetic field at the center of the coil is 0.0750 T ? Express your answer to three significant figures and include the appropriate units. I = 4.64 A Previous Answers Correct Part B At what distance x from the center of the coil, on the axis of the coil, is the magnetic field half its value at the center?

Answer 1

Answer:

I = 4.642 Ampere.

x = 2.145 cm

Explanation:

a) As we know, the magnetic field on the axis of the loop is given as

   $B = \frac{ \mu NIa^{2} }{2(x^{2} + a^{2})^{\frac{3}{2} } }$

where  a = radius of loop        

            x = point on the axis of loop

            N = No of turns of coil

Current in the loop for which the magnetic field at the center is 0.0750 Tesla is given as x = 0

Therefore, the above equation can be rewritten as

               $B_{x} = \frac{ \mu NI}{2a}$

               I = $\frac{2aB}{μN}$

by putting values

I =$\frac{2 X 0.0750 T X 0.028 m}{4\pi X 10^{-7} T.\frac{m}{A} X 720}$

Current in the loop for which the magnetic field at the center is 0.0750 Tesla  = I = 4.642 Ampere

b)

      Now for part b the magnetic field at a distance x from the center is given as

         $B = \frac{ \mu N I a^{2} }{2(x^{2} + a^{2})^{\frac{3}{2} } }$

multiply and divide by a on both sides we get

     $B = \frac{ \mu NI}{2a} X \frac{a^{3} }{(x^{2} + a^{2})^{\frac{3}{2} } }$

   As we know, according to Biot sivorts law,the Magnetic Field at the Center of a circular loop is given as

   $B = \frac{ \mu NI}{2a}$  ( Magnetic field at center) = $B_{c}$

So we got magnetic field at any point x as

$B_{x}$ = $B_{c}$ X  $\frac{a^{3} }{(x^{2} + a^{2})^{\frac{3}{2} } }$

For magnetic field at x is half of the B at center

          $B_{x}$ = $\frac{1}{2}$ $B_{c}$

from the above two equations

$\frac{a^{3} }{(x^{2} + a^{2})^{\frac{3}{2} } }$ = $\frac{1}{2}$

   $(x^{2} + a^{2})^{3} = 4a^{6}$

   $x = \sqrt{4^{\frac{1}{3}} -1 } X a^{}$ 

Putting a = 2.80 cm

We have  x = 2.145 cm Ans

           

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