Question

A closely wound flat circular coil of \[25\] turns of wire has a diameter of \[10\] cm and carries a current of \[4\] amperes. Determine the magnetic flux density at the centre of the coil.
A. \[{1.67910^{ - 5}}T\]
B. \[2.028 \times {10^{ - 4}}T\]
C. \[1.257 \times {10^{ - 3}}T\]
D. \[1.512 \times {10^{ - 6}}T\]

Answer 1

Hint: Magnetic flux density others referred to as magnetic induction assists us to determine the number of magnetic lines of force that pass through a unit area of a given material. The unit of flux density is Tesla which can be represented as T. We can also mention flux density by using the unit, Gauss. The relation between Tesla and Gauss is that one Tesla corresponds to\[10,000\]Gauss. The magnetic field is a vector quantity. This implies that the magnetic field has both magnitude and direction.

Formula used:
$B = \dfrac{{{\mu _o}I}}{{2r}}*n$


Complete step-by-step solution:
Given:
Number of turns n= \[25\]
Diameter of wire d= \[10cm\]
Current I=\[4A\]
${\mu _o} = 4\pi *{10^{ - 7}}$
Radius r$ = \dfrac{d}{2}$
Radius r$ = \dfrac{{10}}{2} = 5cm$
To convert radius unit from centimeter to meter multiply it by ${10^{ - 2}}$
Therefore we get, r=$5cm = 5*{10^{ - 2}}m$
Substituting these values in the equation $B = \dfrac{{{\mu _o}I}}{{2r}}*n$
We get, $B = \dfrac{{4\pi *{{10}^{ - 7}}*4}}{{2*5*{{10}^{ - 2}}}}*25$
Simplifying we get, $B = \dfrac{{4\pi *{{10}^{ - 7}}*100}}{{10*{{10}^{ - 2}}}} = 4\pi *{10^{ - 4}} = 12.56*{10^{ - 4}} = 1.256*{10^{ - 3}}$
The value of magnetic flux is $1.256*{10^{ - 3}}$T.
From the calculation, it is evident that this is not the right option. The value of magnetic flux is$1.256*{10^{ - 3}}$ .
The value of magnetic flux is not\[2.028 \times {10^{ - 4}}\]. Hence this is the wrong option.
The value of magnetic flux$1.256*{10^{ - 3}}$. This is the correct option.
From the solution we have obtained we can determine that this option is wrong.

Note: In this question care must be taken to substitute the value of the radius and not the value of the diameter. Using the equation Radius$ = \dfrac{{diameter}}{2}$ we can change it from diameter to radius. The units of the values must also be taken into consideration. All the values should be in the same unit, preferably in the SI unit format. In this question, it is necessary to convert the unit of radius in centimeters to the meter. The conversion factor \[1{\text{ }}m{\text{ }} = 100{\text{ }}cm\] should be used to convert the unit from centimeter to meter or vice-versa. The value of${\mu _o} = 12.57*{10^{ - 7}}H/m$ but substituting ${\mu _o} = 4\pi *{10^{ - 7}}H/m$will make the calculation process much easier.