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Question: A closely wound flat circular coil of \[25\] turns of wire has a diameter of \[10\] cm and carries a...

A closely wound flat circular coil of 2525 turns of wire has a diameter of 1010 cm and carries a current of 44 amperes. Determine the magnetic flux density at the centre of the coil.
A. 1.67910−5T{1.67910^{ - 5}}T
B. 2.028×10−4T2.028 \times {10^{ - 4}}T
C. 1.257×10−3T1.257 \times {10^{ - 3}}T
D. 1.512×10−6T1.512 \times {10^{ - 6}}T

Explanation

Solution

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 to10,00010,000Gauss. The magnetic field is a vector quantity. This implies that the magnetic field has both magnitude and direction.

Formula used:
B=μoI2r∗nB = \dfrac{{{\mu _o}I}}{{2r}}*n

Complete step-by-step solution:
Given:
Number of turns n= 2525
Diameter of wire d= 10cm10cm
Current I=4A4A
μo=4π∗10−7{\mu _o} = 4\pi *{10^{ - 7}}
Radius r=d2 = \dfrac{d}{2}
Radius r=102=5cm = \dfrac{{10}}{2} = 5cm
To convert radius unit from centimeter to meter multiply it by 10−2{10^{ - 2}}
Therefore we get, r=5cm=5∗10−2m5cm = 5*{10^{ - 2}}m
Substituting these values in the equation B=μoI2r∗nB = \dfrac{{{\mu _o}I}}{{2r}}*n
We get, B=4π∗10−7∗42∗5∗10−2∗25B = \dfrac{{4\pi *{{10}^{ - 7}}*4}}{{2*5*{{10}^{ - 2}}}}*25
Simplifying we get, B=4π∗10−7∗10010∗10−2=4π∗10−4=12.56∗10−4=1.256∗10−3B = \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∗10−31.256*{10^{ - 3}}T.
From the calculation, it is evident that this is not the right option. The value of magnetic flux is1.256∗10−31.256*{10^{ - 3}} .
The value of magnetic flux is not2.028×10−42.028 \times {10^{ - 4}}. Hence this is the wrong option.
The value of magnetic flux1.256∗10−31.256*{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=diameter2 = \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 m =100 cm1{\text{ }}m{\text{ }} = 100{\text{ }}cm should be used to convert the unit from centimeter to meter or vice-versa. The value ofμo=12.57∗10−7H/m{\mu _o} = 12.57*{10^{ - 7}}H/m but substituting μo=4π∗10−7H/m{\mu _o} = 4\pi *{10^{ - 7}}H/mwill make the calculation process much easier.